KILLED
proof of input_ER9hDJlpIa.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 9 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 6148 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 3 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 238 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 184 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 49 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 9 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 878 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 388 ms]
        (56) CpxRNTS
        (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 827 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 260 ms]
        (62) CpxRNTS
        (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
        (65) IntTrsBoundProof [UPPER BOUND(ID), 158.0 s]
        (66) CpxRNTS
        (67) IntTrsBoundProof [UPPER BOUND(ID), 34.7 s]
        (68) CpxRNTS
    (69) CompletionProof [UPPER BOUND(ID), 0 ms]
    (70) CpxTypedWeightedCompleteTrs
        (71) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (72) CpxRNTS
(73) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (80) CdtProblem
    (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (108) CdtProblem
    (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 410 ms]
    (126) CdtProblem
    (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 451 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (146) CdtProblem
    (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 15 ms]
    (148) CdtProblem
    (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms]
    (150) CdtProblem
    (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (188) CdtProblem
    (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms]
    (190) CdtProblem
    (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   head(add(n, x)) -> n
   tail(add(n, x)) -> x
   isempty(nil) -> true
   isempty(add(n, x)) -> false
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x)))
   if_qs(true, x, n, y) -> nil
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1]
   if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1]
   if_low(false, n, add(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1]
   if_high(true, n, add(m, x)) -> high(n, x) [1]
   if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1]
   head(add(n, x)) -> n [1]
   tail(add(n, x)) -> x [1]
   isempty(nil) -> true [1]
   isempty(add(n, x)) -> false [1]
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) [1]
   if_qs(true, x, n, y) -> nil [1]
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1]
   if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1]
   if_low(false, n, add(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1]
   if_high(true, n, add(m, x)) -> high(n, x) [1]
   if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1]
   head(add(n, x)) -> n [1]
   tail(add(n, x)) -> x [1]
   isempty(nil) -> true [1]
   isempty(add(n, x)) -> false [1]
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) [1]
   if_qs(true, x, n, y) -> nil [1]
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y))) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0:s -> nil:add -> nil:add
low :: 0:s -> nil:add -> nil:add
if_low :: true:false -> 0:s -> nil:add -> nil:add
high :: 0:s -> nil:add -> nil:add
if_high :: true:false -> 0:s -> nil:add -> nil:add
head :: nil:add -> 0:s
tail :: nil:add -> nil:add
isempty :: nil:add -> true:false
quicksort :: nil:add -> nil:add
if_qs :: true:false -> nil:add -> 0:s -> nil:add -> nil:add

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   le_2
   isempty_1
   low_2
   head_1
   tail_1
   high_2
   quicksort_1
   if_high_3
   if_qs_4
   if_low_3
   app_2

Due to the following rules being added:

   head(v0) -> 0 [0]
   tail(v0) -> nil [0]
   if_high(v0, v1, v2) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]

And the following fresh constants: none

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1]
   if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1]
   if_low(false, n, add(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1]
   if_high(true, n, add(m, x)) -> high(n, x) [1]
   if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1]
   head(add(n, x)) -> n [1]
   tail(add(n, x)) -> x [1]
   isempty(nil) -> true [1]
   isempty(add(n, x)) -> false [1]
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) [1]
   if_qs(true, x, n, y) -> nil [1]
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y))) [1]
   head(v0) -> 0 [0]
   tail(v0) -> nil [0]
   if_high(v0, v1, v2) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0:s -> nil:add -> nil:add
low :: 0:s -> nil:add -> nil:add
if_low :: true:false -> 0:s -> nil:add -> nil:add
high :: 0:s -> nil:add -> nil:add
if_high :: true:false -> 0:s -> nil:add -> nil:add
head :: nil:add -> 0:s
tail :: nil:add -> nil:add
isempty :: nil:add -> true:false
quicksort :: nil:add -> nil:add
if_qs :: true:false -> nil:add -> 0:s -> nil:add -> nil:add

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, add(0, x)) -> if_low(true, n, add(0, x)) [2]
   low(0, add(s(x'), x)) -> if_low(false, 0, add(s(x'), x)) [2]
   low(s(y'), add(s(x''), x)) -> if_low(le(x'', y'), s(y'), add(s(x''), x)) [2]
   if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1]
   if_low(false, n, add(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, add(0, x)) -> if_high(true, n, add(0, x)) [2]
   high(0, add(s(x1), x)) -> if_high(false, 0, add(s(x1), x)) [2]
   high(s(y''), add(s(x2), x)) -> if_high(le(x2, y''), s(y''), add(s(x2), x)) [2]
   if_high(true, n, add(m, x)) -> high(n, x) [1]
   if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1]
   head(add(n, x)) -> n [1]
   tail(add(n, x)) -> x [1]
   isempty(nil) -> true [1]
   isempty(add(n, x)) -> false [1]
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) [1]
   if_qs(true, x, n, y) -> nil [1]
   if_qs(false, x, n, y) -> app(if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))), add(n, if_qs(isempty(y), low(head(y), tail(y)), head(y), high(head(y), tail(y))))) [3]
   head(v0) -> 0 [0]
   tail(v0) -> nil [0]
   if_high(v0, v1, v2) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0:s -> nil:add -> nil:add
low :: 0:s -> nil:add -> nil:add
if_low :: true:false -> 0:s -> nil:add -> nil:add
high :: 0:s -> nil:add -> nil:add
if_high :: true:false -> 0:s -> nil:add -> nil:add
head :: nil:add -> 0:s
tail :: nil:add -> nil:add
isempty :: nil:add -> true:false
quicksort :: nil:add -> nil:add
if_qs :: true:false -> nil:add -> 0:s -> nil:add -> nil:add

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 1
   false => 0
   nil => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   high(z, z') -{ 2 }-> if_high(le(x2, y''), 1 + y'', 1 + (1 + x2) + x) :|: z = 1 + y'', z' = 1 + (1 + x2) + x, x >= 0, y'' >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, n, 1 + 0 + x) :|: n >= 0, z = n, x >= 0, z' = 1 + 0 + x
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_qs(z, z', z'', z1) -{ 3 }-> app(if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))), 1 + n + if_qs(isempty(y), low(head(y), tail(y)), head(y), high(head(y), tail(y)))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: n >= 0, z1 = y, z' = x, z'' = n, z = 1, x >= 0, y >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', y'), 1 + y', 1 + (1 + x'') + x) :|: x >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, n, 1 + 0 + x) :|: n >= 0, z = n, x >= 0, z' = 1 + 0 + x
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   quicksort(z) -{ 1 }-> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) :|: x >= 0, z = x
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


----------------------------------------

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   isempty(z) -{ 1 }-> 1 :|: z = 0

----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   high(z, z') -{ 2 }-> if_high(le(x2, y''), 1 + y'', 1 + (1 + x2) + x) :|: z = 1 + y'', z' = 1 + (1 + x2) + x, x >= 0, y'' >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, n, 1 + 0 + x) :|: n >= 0, z = n, x >= 0, z' = 1 + 0 + x
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(n, x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(1, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, y = v0'', v01 >= 0, y = v01, x = 0, y = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), x'), head(y), high(head(y), x''))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x' >= 0, y = 1 + n' + x', n'' >= 0, x'' >= 0, y = 1 + n'' + x'', x = 0, n1 >= 0, x1 >= 0, y = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), x'), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x' >= 0, y = 1 + n' + x', v0'' >= 0, y = v0'', x = 0, n'' >= 0, x'' >= 0, y = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), x'))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, y = v0'', n' >= 0, x' >= 0, y = 1 + n' + x', x = 0, n'' >= 0, x'' >= 0, y = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, y = v0'', v01 >= 0, y = v01, x = 0, n' >= 0, x' >= 0, y = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), x'')), 1 + n + if_qs(1, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', v0 >= 0, y = v0, v0' >= 0, y = v0', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, y = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), x'')), 1 + n + if_qs(0, low(head(y), x1), head(y), high(head(y), x2))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2, n3 >= 0, x3 >= 0, x = 1 + n3 + x3, n4 >= 0, x4 >= 0, y = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), x'')), 1 + n + if_qs(0, low(head(y), x1), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1, v0 >= 0, y = v0, n2 >= 0, x2 >= 0, x = 1 + n2 + x2, n3 >= 0, x3 >= 0, y = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), x'')), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), x1))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', v0 >= 0, y = v0, n1 >= 0, x1 >= 0, y = 1 + n1 + x1, n2 >= 0, x2 >= 0, x = 1 + n2 + x2, n3 >= 0, x3 >= 0, y = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), x'')), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', v0 >= 0, y = v0, v0' >= 0, y = v0', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), 0)), 1 + n + if_qs(1, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', v0 >= 0, x = v0, v0' >= 0, y = v0', v0'' >= 0, y = v0'', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', y = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), x''), head(y), high(head(y), x1))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', v0 >= 0, x = v0, n'' >= 0, x'' >= 0, y = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1, n2 >= 0, x2 >= 0, x = 1 + n2 + x2, n3 >= 0, x3 >= 0, y = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), x''), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', v0 >= 0, x = v0, n'' >= 0, x'' >= 0, y = 1 + n'' + x'', v0' >= 0, y = v0', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), x''))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', v0 >= 0, x = v0, v0' >= 0, y = v0', n'' >= 0, x'' >= 0, y = 1 + n'' + x'', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(x), x'), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, n' >= 0, x' >= 0, x = 1 + n' + x', v0 >= 0, x = v0, v0' >= 0, y = v0', v0'' >= 0, y = v0'', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), x')), 1 + n + if_qs(1, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, n' >= 0, x' >= 0, x = 1 + n' + x', v0' >= 0, y = v0', v0'' >= 0, y = v0'', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', y = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), x')), 1 + n + if_qs(0, low(head(y), x''), head(y), high(head(y), x1))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, y = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1, n2 >= 0, x2 >= 0, x = 1 + n2 + x2, n3 >= 0, x3 >= 0, y = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), x')), 1 + n + if_qs(0, low(head(y), x''), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, y = 1 + n'' + x'', v0' >= 0, y = v0', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), x')), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), x''))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, n' >= 0, x' >= 0, x = 1 + n' + x', v0' >= 0, y = v0', n'' >= 0, x'' >= 0, y = 1 + n'' + x'', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), x')), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, n' >= 0, x' >= 0, x = 1 + n' + x', v0' >= 0, y = v0', v0'' >= 0, y = v0'', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(1, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, y = v0'', v01 >= 0, y = v01, n' >= 0, x' >= 0, x = 1 + n' + x', y = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), x'), head(y), high(head(y), x''))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x' >= 0, y = 1 + n' + x', n'' >= 0, x'' >= 0, y = 1 + n'' + x'', n1 >= 0, x1 >= 0, x = 1 + n1 + x1, n2 >= 0, x2 >= 0, y = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), x'), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x' >= 0, y = 1 + n' + x', v0'' >= 0, y = v0'', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), x'))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, y = v0'', n' >= 0, x' >= 0, y = 1 + n' + x', n'' >= 0, x'' >= 0, x = 1 + n'' + x'', n1 >= 0, x1 >= 0, y = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(x), 0), head(x), high(head(x), 0)), 1 + n + if_qs(0, low(head(y), 0), head(y), high(head(y), 0))) :|: n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0, z = 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, y = v0'', v01 >= 0, y = v01, n' >= 0, x' >= 0, x = 1 + n' + x', n'' >= 0, x'' >= 0, y = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: n >= 0, z1 = y, z' = x, z'' = n, z = 1, x >= 0, y >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', y'), 1 + y', 1 + (1 + x'') + x) :|: x >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, n, 1 + 0 + x) :|: n >= 0, z = n, x >= 0, z' = 1 + 0 + x
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, v02 >= 0, x = v02, x = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0'' >= 0, x = v0'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0 >= 0, x = v0, n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2, n2 >= 0, x3 >= 0, x = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0 >= 0, x = v0, n'' >= 0, x1 >= 0, x = 1 + n'' + x1, v0' >= 0, x = v0', n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2, n2 >= 0, x3 >= 0, x = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, v0' >= 0, x = v0', n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2, n2 >= 0, x3 >= 0, x = 1 + n2 + x3, n3 >= 0, x4 >= 0, x = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2, v0 >= 0, x = v0, n2 >= 0, x3 >= 0, x = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, v0 >= 0, x = v0, n1 >= 0, x2 >= 0, x = 1 + n1 + x2, n2 >= 0, x3 >= 0, x = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, v0 >= 0, x = v0, v0' >= 0, x = v0', n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0 >= 0, x = v0, v0' >= 0, x = v0', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, n' >= 0, x'' >= 0, x = 1 + n' + x'', v0' >= 0, x = v0', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, n' >= 0, x'' >= 0, x = 1 + n' + x'', v0' >= 0, x = v0', v0'' >= 0, x = v0'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: x >= 0, z = x, n >= 0, x' >= 0, x = 1 + n + x', v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, n' >= 0, x'' >= 0, x = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x' >= 0, x = 1 + n + x', v01 >= 0, x = v01, n' >= 0, x'' >= 0, x = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', v0' >= 0, x = v0', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', v0' >= 0, x = v0', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0'' >= 0, x = v0'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0'' >= 0, x = v0'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2, n2 >= 0, x3 >= 0, x = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, v0' >= 0, x = v0', n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0' >= 0, x = v0', n'' >= 0, x1 >= 0, x = 1 + n'' + x1, n1 >= 0, x2 >= 0, x = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x'', v0' >= 0, x = v0', v0'' >= 0, x = v0'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', v0' >= 0, x = v0', v0'' >= 0, x = v0'', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, n >= 0, x' >= 0, x = 1 + n + x', v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, n' >= 0, x'' >= 0, x = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x' >= 0, x = 1 + n + x', v0'' >= 0, x = v0'', n' >= 0, x'' >= 0, x = 1 + n' + x'', n'' >= 0, x1 >= 0, x = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x' >= 0, x = 1 + n + x', v0'' >= 0, x = v0'', v01 >= 0, x = v01, n' >= 0, x'' >= 0, x = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, n >= 0, x' >= 0, x = 1 + n + x', n' >= 0, x'' >= 0, x = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: x >= 0, z = x, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, v02 >= 0, x = v02, n >= 0, x' >= 0, x = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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(17) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 2 }-> if_high(le(x2, z - 1), 1 + (z - 1), 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + x) :|: x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0


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(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { le }
   { tail }
   { isempty }
   { head }
   { app }
   { high, if_high }
   { low, if_low }
   { if_qs }
   { quicksort }

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 2 }-> if_high(le(x2, z - 1), 1 + (z - 1), 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + x) :|: x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}

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(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 2 }-> if_high(le(x2, z - 1), 1 + (z - 1), 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + x) :|: x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}

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(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 2 }-> if_high(le(x2, z - 1), 1 + (z - 1), 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + x) :|: x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 2 }-> if_high(le(x2, z - 1), 1 + (z - 1), 1 + (1 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 2 }-> if_low(le(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + x) :|: x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: ?, size: O(n^1) [z]

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]

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(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(34)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]

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(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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(36)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isempty}, {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: ?, size: O(1) [1]

----------------------------------------

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]

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(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: ?, size: O(n^1) [z]

----------------------------------------

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(1, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 9 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), x2))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), x1), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), x'')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), x'), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 8 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), x1))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), x''), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), x')), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(1, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 7 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), x''))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), x'), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 6 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), x'))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 5 }-> app(if_qs(0, low(head(z'), 0), head(z'), high(head(z'), 0)), 1 + z'' + if_qs(0, low(head(z1), 0), head(z1), high(head(z1), 0))) :|: z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]

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(45) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s109, 0), s110, high(s111, 0)), 1 + z'' + if_qs(0, low(s112, x'), s113, high(s114, 0))) :|: s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s121, 0), s122, high(s123, 0)), 1 + z'' + if_qs(0, low(s124, 0), s125, high(s126, x'))) :|: s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s139, 0), s140, high(s141, 0)), 1 + z'' + if_qs(0, low(s142, 0), s143, high(s144, 0))) :|: s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s145, 0), s146, high(s147, 0)), 1 + z'' + if_qs(1, low(s148, 0), s149, high(s150, 0))) :|: s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(1, low(s97, 0), s98, high(s99, 0)), 1 + z'' + if_qs(0, low(s100, x'), s101, high(s102, x''))) :|: s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 15 }-> app(if_qs(0, low(s1, x'), s2, high(s3, x'')), 1 + z'' + if_qs(0, low(s4, x1), s5, high(s6, x2))) :|: s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s103, 0), s104, high(s105, 0)), 1 + z'' + if_qs(0, low(s106, x'), s107, high(s108, 0))) :|: s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s115, 0), s116, high(s117, 0)), 1 + z'' + if_qs(0, low(s118, 0), s119, high(s120, x'))) :|: s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s127, 0), s128, high(s129, 0)), 1 + z'' + if_qs(0, low(s130, 0), s131, high(s132, 0))) :|: s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s13, x'), s14, high(s15, x'')), 1 + z'' + if_qs(0, low(s16, 0), s17, high(s18, x1))) :|: s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s133, 0), s134, high(s135, 0)), 1 + z'' + if_qs(1, low(s136, 0), s137, high(s138, 0))) :|: s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s19, x'), s20, high(s21, x'')), 1 + z'' + if_qs(0, low(s22, 0), s23, high(s24, 0))) :|: s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s25, x'), s26, high(s27, x'')), 1 + z'' + if_qs(1, low(s28, 0), s29, high(s30, 0))) :|: s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s31, x'), s32, high(s33, 0)), 1 + z'' + if_qs(0, low(s34, x''), s35, high(s36, x1))) :|: s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s37, x'), s38, high(s39, 0)), 1 + z'' + if_qs(0, low(s40, x''), s41, high(s42, 0))) :|: s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s43, x'), s44, high(s45, 0)), 1 + z'' + if_qs(0, low(s46, 0), s47, high(s48, x''))) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s49, x'), s50, high(s51, 0)), 1 + z'' + if_qs(0, low(s52, 0), s53, high(s54, 0))) :|: s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s55, x'), s56, high(s57, 0)), 1 + z'' + if_qs(1, low(s58, 0), s59, high(s60, 0))) :|: s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s61, 0), s62, high(s63, x')), 1 + z'' + if_qs(0, low(s64, x''), s65, high(s66, x1))) :|: s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s67, 0), s68, high(s69, x')), 1 + z'' + if_qs(0, low(s70, x''), s71, high(s72, 0))) :|: s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s7, x'), s8, high(s9, x'')), 1 + z'' + if_qs(0, low(s10, x1), s11, high(s12, 0))) :|: s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s73, 0), s74, high(s75, x')), 1 + z'' + if_qs(0, low(s76, 0), s77, high(s78, x''))) :|: s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s79, 0), s80, high(s81, x')), 1 + z'' + if_qs(0, low(s82, 0), s83, high(s84, 0))) :|: s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s85, 0), s86, high(s87, x')), 1 + z'' + if_qs(1, low(s88, 0), s89, high(s90, 0))) :|: s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s91, 0), s92, high(s93, 0)), 1 + z'' + if_qs(0, low(s94, x'), s95, high(s96, x''))) :|: s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]

----------------------------------------

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s109, 0), s110, high(s111, 0)), 1 + z'' + if_qs(0, low(s112, x'), s113, high(s114, 0))) :|: s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s121, 0), s122, high(s123, 0)), 1 + z'' + if_qs(0, low(s124, 0), s125, high(s126, x'))) :|: s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s139, 0), s140, high(s141, 0)), 1 + z'' + if_qs(0, low(s142, 0), s143, high(s144, 0))) :|: s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s145, 0), s146, high(s147, 0)), 1 + z'' + if_qs(1, low(s148, 0), s149, high(s150, 0))) :|: s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(1, low(s97, 0), s98, high(s99, 0)), 1 + z'' + if_qs(0, low(s100, x'), s101, high(s102, x''))) :|: s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 15 }-> app(if_qs(0, low(s1, x'), s2, high(s3, x'')), 1 + z'' + if_qs(0, low(s4, x1), s5, high(s6, x2))) :|: s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s103, 0), s104, high(s105, 0)), 1 + z'' + if_qs(0, low(s106, x'), s107, high(s108, 0))) :|: s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s115, 0), s116, high(s117, 0)), 1 + z'' + if_qs(0, low(s118, 0), s119, high(s120, x'))) :|: s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s127, 0), s128, high(s129, 0)), 1 + z'' + if_qs(0, low(s130, 0), s131, high(s132, 0))) :|: s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s13, x'), s14, high(s15, x'')), 1 + z'' + if_qs(0, low(s16, 0), s17, high(s18, x1))) :|: s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s133, 0), s134, high(s135, 0)), 1 + z'' + if_qs(1, low(s136, 0), s137, high(s138, 0))) :|: s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s19, x'), s20, high(s21, x'')), 1 + z'' + if_qs(0, low(s22, 0), s23, high(s24, 0))) :|: s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s25, x'), s26, high(s27, x'')), 1 + z'' + if_qs(1, low(s28, 0), s29, high(s30, 0))) :|: s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s31, x'), s32, high(s33, 0)), 1 + z'' + if_qs(0, low(s34, x''), s35, high(s36, x1))) :|: s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s37, x'), s38, high(s39, 0)), 1 + z'' + if_qs(0, low(s40, x''), s41, high(s42, 0))) :|: s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s43, x'), s44, high(s45, 0)), 1 + z'' + if_qs(0, low(s46, 0), s47, high(s48, x''))) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s49, x'), s50, high(s51, 0)), 1 + z'' + if_qs(0, low(s52, 0), s53, high(s54, 0))) :|: s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s55, x'), s56, high(s57, 0)), 1 + z'' + if_qs(1, low(s58, 0), s59, high(s60, 0))) :|: s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s61, 0), s62, high(s63, x')), 1 + z'' + if_qs(0, low(s64, x''), s65, high(s66, x1))) :|: s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s67, 0), s68, high(s69, x')), 1 + z'' + if_qs(0, low(s70, x''), s71, high(s72, 0))) :|: s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s7, x'), s8, high(s9, x'')), 1 + z'' + if_qs(0, low(s10, x1), s11, high(s12, 0))) :|: s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s73, 0), s74, high(s75, x')), 1 + z'' + if_qs(0, low(s76, 0), s77, high(s78, x''))) :|: s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s79, 0), s80, high(s81, x')), 1 + z'' + if_qs(0, low(s82, 0), s83, high(s84, 0))) :|: s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s85, 0), s86, high(s87, x')), 1 + z'' + if_qs(1, low(s88, 0), s89, high(s90, 0))) :|: s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s91, 0), s92, high(s93, 0)), 1 + z'' + if_qs(0, low(s94, x'), s95, high(s96, x''))) :|: s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {app}, {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: ?, size: O(n^1) [z + z']

----------------------------------------

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + n + app(x, z') :|: n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s109, 0), s110, high(s111, 0)), 1 + z'' + if_qs(0, low(s112, x'), s113, high(s114, 0))) :|: s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s121, 0), s122, high(s123, 0)), 1 + z'' + if_qs(0, low(s124, 0), s125, high(s126, x'))) :|: s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s139, 0), s140, high(s141, 0)), 1 + z'' + if_qs(0, low(s142, 0), s143, high(s144, 0))) :|: s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s145, 0), s146, high(s147, 0)), 1 + z'' + if_qs(1, low(s148, 0), s149, high(s150, 0))) :|: s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(1, low(s97, 0), s98, high(s99, 0)), 1 + z'' + if_qs(0, low(s100, x'), s101, high(s102, x''))) :|: s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 15 }-> app(if_qs(0, low(s1, x'), s2, high(s3, x'')), 1 + z'' + if_qs(0, low(s4, x1), s5, high(s6, x2))) :|: s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s103, 0), s104, high(s105, 0)), 1 + z'' + if_qs(0, low(s106, x'), s107, high(s108, 0))) :|: s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s115, 0), s116, high(s117, 0)), 1 + z'' + if_qs(0, low(s118, 0), s119, high(s120, x'))) :|: s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s127, 0), s128, high(s129, 0)), 1 + z'' + if_qs(0, low(s130, 0), s131, high(s132, 0))) :|: s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s13, x'), s14, high(s15, x'')), 1 + z'' + if_qs(0, low(s16, 0), s17, high(s18, x1))) :|: s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s133, 0), s134, high(s135, 0)), 1 + z'' + if_qs(1, low(s136, 0), s137, high(s138, 0))) :|: s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s19, x'), s20, high(s21, x'')), 1 + z'' + if_qs(0, low(s22, 0), s23, high(s24, 0))) :|: s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s25, x'), s26, high(s27, x'')), 1 + z'' + if_qs(1, low(s28, 0), s29, high(s30, 0))) :|: s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s31, x'), s32, high(s33, 0)), 1 + z'' + if_qs(0, low(s34, x''), s35, high(s36, x1))) :|: s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s37, x'), s38, high(s39, 0)), 1 + z'' + if_qs(0, low(s40, x''), s41, high(s42, 0))) :|: s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s43, x'), s44, high(s45, 0)), 1 + z'' + if_qs(0, low(s46, 0), s47, high(s48, x''))) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s49, x'), s50, high(s51, 0)), 1 + z'' + if_qs(0, low(s52, 0), s53, high(s54, 0))) :|: s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s55, x'), s56, high(s57, 0)), 1 + z'' + if_qs(1, low(s58, 0), s59, high(s60, 0))) :|: s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s61, 0), s62, high(s63, x')), 1 + z'' + if_qs(0, low(s64, x''), s65, high(s66, x1))) :|: s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s67, 0), s68, high(s69, x')), 1 + z'' + if_qs(0, low(s70, x''), s71, high(s72, 0))) :|: s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s7, x'), s8, high(s9, x'')), 1 + z'' + if_qs(0, low(s10, x1), s11, high(s12, 0))) :|: s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s73, 0), s74, high(s75, x')), 1 + z'' + if_qs(0, low(s76, 0), s77, high(s78, x''))) :|: s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s79, 0), s80, high(s81, x')), 1 + z'' + if_qs(0, low(s82, 0), s83, high(s84, 0))) :|: s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s85, 0), s86, high(s87, x')), 1 + z'' + if_qs(1, low(s88, 0), s89, high(s90, 0))) :|: s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s91, 0), s92, high(s93, 0)), 1 + z'' + if_qs(0, low(s94, x'), s95, high(s96, x''))) :|: s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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(51) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(52)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s109, 0), s110, high(s111, 0)), 1 + z'' + if_qs(0, low(s112, x'), s113, high(s114, 0))) :|: s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s121, 0), s122, high(s123, 0)), 1 + z'' + if_qs(0, low(s124, 0), s125, high(s126, x'))) :|: s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s139, 0), s140, high(s141, 0)), 1 + z'' + if_qs(0, low(s142, 0), s143, high(s144, 0))) :|: s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s145, 0), s146, high(s147, 0)), 1 + z'' + if_qs(1, low(s148, 0), s149, high(s150, 0))) :|: s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(1, low(s97, 0), s98, high(s99, 0)), 1 + z'' + if_qs(0, low(s100, x'), s101, high(s102, x''))) :|: s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 15 }-> app(if_qs(0, low(s1, x'), s2, high(s3, x'')), 1 + z'' + if_qs(0, low(s4, x1), s5, high(s6, x2))) :|: s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s103, 0), s104, high(s105, 0)), 1 + z'' + if_qs(0, low(s106, x'), s107, high(s108, 0))) :|: s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s115, 0), s116, high(s117, 0)), 1 + z'' + if_qs(0, low(s118, 0), s119, high(s120, x'))) :|: s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s127, 0), s128, high(s129, 0)), 1 + z'' + if_qs(0, low(s130, 0), s131, high(s132, 0))) :|: s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s13, x'), s14, high(s15, x'')), 1 + z'' + if_qs(0, low(s16, 0), s17, high(s18, x1))) :|: s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s133, 0), s134, high(s135, 0)), 1 + z'' + if_qs(1, low(s136, 0), s137, high(s138, 0))) :|: s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s19, x'), s20, high(s21, x'')), 1 + z'' + if_qs(0, low(s22, 0), s23, high(s24, 0))) :|: s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s25, x'), s26, high(s27, x'')), 1 + z'' + if_qs(1, low(s28, 0), s29, high(s30, 0))) :|: s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s31, x'), s32, high(s33, 0)), 1 + z'' + if_qs(0, low(s34, x''), s35, high(s36, x1))) :|: s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s37, x'), s38, high(s39, 0)), 1 + z'' + if_qs(0, low(s40, x''), s41, high(s42, 0))) :|: s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s43, x'), s44, high(s45, 0)), 1 + z'' + if_qs(0, low(s46, 0), s47, high(s48, x''))) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s49, x'), s50, high(s51, 0)), 1 + z'' + if_qs(0, low(s52, 0), s53, high(s54, 0))) :|: s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s55, x'), s56, high(s57, 0)), 1 + z'' + if_qs(1, low(s58, 0), s59, high(s60, 0))) :|: s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s61, 0), s62, high(s63, x')), 1 + z'' + if_qs(0, low(s64, x''), s65, high(s66, x1))) :|: s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s67, 0), s68, high(s69, x')), 1 + z'' + if_qs(0, low(s70, x''), s71, high(s72, 0))) :|: s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s7, x'), s8, high(s9, x'')), 1 + z'' + if_qs(0, low(s10, x1), s11, high(s12, 0))) :|: s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s73, 0), s74, high(s75, x')), 1 + z'' + if_qs(0, low(s76, 0), s77, high(s78, x''))) :|: s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s79, 0), s80, high(s81, x')), 1 + z'' + if_qs(0, low(s82, 0), s83, high(s84, 0))) :|: s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s85, 0), s86, high(s87, x')), 1 + z'' + if_qs(1, low(s88, 0), s89, high(s90, 0))) :|: s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s91, 0), s92, high(s93, 0)), 1 + z'' + if_qs(0, low(s94, x'), s95, high(s96, x''))) :|: s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

----------------------------------------

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: high
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

Computed SIZE bound using KoAT for: if_high
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z''

----------------------------------------

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s109, 0), s110, high(s111, 0)), 1 + z'' + if_qs(0, low(s112, x'), s113, high(s114, 0))) :|: s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s121, 0), s122, high(s123, 0)), 1 + z'' + if_qs(0, low(s124, 0), s125, high(s126, x'))) :|: s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s139, 0), s140, high(s141, 0)), 1 + z'' + if_qs(0, low(s142, 0), s143, high(s144, 0))) :|: s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s145, 0), s146, high(s147, 0)), 1 + z'' + if_qs(1, low(s148, 0), s149, high(s150, 0))) :|: s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(1, low(s97, 0), s98, high(s99, 0)), 1 + z'' + if_qs(0, low(s100, x'), s101, high(s102, x''))) :|: s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 15 }-> app(if_qs(0, low(s1, x'), s2, high(s3, x'')), 1 + z'' + if_qs(0, low(s4, x1), s5, high(s6, x2))) :|: s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s103, 0), s104, high(s105, 0)), 1 + z'' + if_qs(0, low(s106, x'), s107, high(s108, 0))) :|: s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s115, 0), s116, high(s117, 0)), 1 + z'' + if_qs(0, low(s118, 0), s119, high(s120, x'))) :|: s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s127, 0), s128, high(s129, 0)), 1 + z'' + if_qs(0, low(s130, 0), s131, high(s132, 0))) :|: s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s13, x'), s14, high(s15, x'')), 1 + z'' + if_qs(0, low(s16, 0), s17, high(s18, x1))) :|: s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s133, 0), s134, high(s135, 0)), 1 + z'' + if_qs(1, low(s136, 0), s137, high(s138, 0))) :|: s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s19, x'), s20, high(s21, x'')), 1 + z'' + if_qs(0, low(s22, 0), s23, high(s24, 0))) :|: s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s25, x'), s26, high(s27, x'')), 1 + z'' + if_qs(1, low(s28, 0), s29, high(s30, 0))) :|: s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s31, x'), s32, high(s33, 0)), 1 + z'' + if_qs(0, low(s34, x''), s35, high(s36, x1))) :|: s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s37, x'), s38, high(s39, 0)), 1 + z'' + if_qs(0, low(s40, x''), s41, high(s42, 0))) :|: s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s43, x'), s44, high(s45, 0)), 1 + z'' + if_qs(0, low(s46, 0), s47, high(s48, x''))) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s49, x'), s50, high(s51, 0)), 1 + z'' + if_qs(0, low(s52, 0), s53, high(s54, 0))) :|: s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s55, x'), s56, high(s57, 0)), 1 + z'' + if_qs(1, low(s58, 0), s59, high(s60, 0))) :|: s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s61, 0), s62, high(s63, x')), 1 + z'' + if_qs(0, low(s64, x''), s65, high(s66, x1))) :|: s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s67, 0), s68, high(s69, x')), 1 + z'' + if_qs(0, low(s70, x''), s71, high(s72, 0))) :|: s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s7, x'), s8, high(s9, x'')), 1 + z'' + if_qs(0, low(s10, x1), s11, high(s12, 0))) :|: s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s73, 0), s74, high(s75, x')), 1 + z'' + if_qs(0, low(s76, 0), s77, high(s78, x''))) :|: s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s79, 0), s80, high(s81, x')), 1 + z'' + if_qs(0, low(s82, 0), s83, high(s84, 0))) :|: s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s85, 0), s86, high(s87, x')), 1 + z'' + if_qs(1, low(s88, 0), s89, high(s90, 0))) :|: s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s91, 0), s92, high(s93, 0)), 1 + z'' + if_qs(0, low(s94, x'), s95, high(s96, x''))) :|: s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {high,if_high}, {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: ?, size: O(n^1) [z']
  if_high: runtime: ?, size: O(n^1) [z'']

----------------------------------------

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: high
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 11 + 3*z + z*z' + 4*z'

Computed RUNTIME bound using KoAT for: if_high
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z'' + 8*z''

----------------------------------------

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 3 + z }-> if_high(s'', 1 + (z - 1), 1 + (1 + x2) + x) :|: s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 2 }-> if_high(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   high(z, z') -{ 2 }-> if_high(0, 0, 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 1 }-> high(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s109, 0), s110, high(s111, 0)), 1 + z'' + if_qs(0, low(s112, x'), s113, high(s114, 0))) :|: s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(1, low(s121, 0), s122, high(s123, 0)), 1 + z'' + if_qs(0, low(s124, 0), s125, high(s126, x'))) :|: s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s139, 0), s140, high(s141, 0)), 1 + z'' + if_qs(0, low(s142, 0), s143, high(s144, 0))) :|: s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(1, low(s145, 0), s146, high(s147, 0)), 1 + z'' + if_qs(1, low(s148, 0), s149, high(s150, 0))) :|: s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(1, low(s97, 0), s98, high(s99, 0)), 1 + z'' + if_qs(0, low(s100, x'), s101, high(s102, x''))) :|: s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 15 }-> app(if_qs(0, low(s1, x'), s2, high(s3, x'')), 1 + z'' + if_qs(0, low(s4, x1), s5, high(s6, x2))) :|: s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s103, 0), s104, high(s105, 0)), 1 + z'' + if_qs(0, low(s106, x'), s107, high(s108, 0))) :|: s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s115, 0), s116, high(s117, 0)), 1 + z'' + if_qs(0, low(s118, 0), s119, high(s120, x'))) :|: s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s127, 0), s128, high(s129, 0)), 1 + z'' + if_qs(0, low(s130, 0), s131, high(s132, 0))) :|: s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s13, x'), s14, high(s15, x'')), 1 + z'' + if_qs(0, low(s16, 0), s17, high(s18, x1))) :|: s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 11 }-> app(if_qs(0, low(s133, 0), s134, high(s135, 0)), 1 + z'' + if_qs(1, low(s136, 0), s137, high(s138, 0))) :|: s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s19, x'), s20, high(s21, x'')), 1 + z'' + if_qs(0, low(s22, 0), s23, high(s24, 0))) :|: s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s25, x'), s26, high(s27, x'')), 1 + z'' + if_qs(1, low(s28, 0), s29, high(s30, 0))) :|: s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s31, x'), s32, high(s33, 0)), 1 + z'' + if_qs(0, low(s34, x''), s35, high(s36, x1))) :|: s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s37, x'), s38, high(s39, 0)), 1 + z'' + if_qs(0, low(s40, x''), s41, high(s42, 0))) :|: s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s43, x'), s44, high(s45, 0)), 1 + z'' + if_qs(0, low(s46, 0), s47, high(s48, x''))) :|: s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s49, x'), s50, high(s51, 0)), 1 + z'' + if_qs(0, low(s52, 0), s53, high(s54, 0))) :|: s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s55, x'), s56, high(s57, 0)), 1 + z'' + if_qs(1, low(s58, 0), s59, high(s60, 0))) :|: s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s61, 0), s62, high(s63, x')), 1 + z'' + if_qs(0, low(s64, x''), s65, high(s66, x1))) :|: s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s67, 0), s68, high(s69, x')), 1 + z'' + if_qs(0, low(s70, x''), s71, high(s72, 0))) :|: s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 14 }-> app(if_qs(0, low(s7, x'), s8, high(s9, x'')), 1 + z'' + if_qs(0, low(s10, x1), s11, high(s12, 0))) :|: s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s73, 0), s74, high(s75, x')), 1 + z'' + if_qs(0, low(s76, 0), s77, high(s78, x''))) :|: s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s79, 0), s80, high(s81, x')), 1 + z'' + if_qs(0, low(s82, 0), s83, high(s84, 0))) :|: s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 12 }-> app(if_qs(0, low(s85, 0), s86, high(s87, x')), 1 + z'' + if_qs(1, low(s88, 0), s89, high(s90, 0))) :|: s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 13 }-> app(if_qs(0, low(s91, 0), s92, high(s93, 0)), 1 + z'' + if_qs(0, low(s94, x'), s95, high(s96, x''))) :|: s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 2 }-> if_qs(1, low(0, 0), 0, high(0, 0)) :|: z >= 0, z = 0
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x''), 0, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, x''), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), n', high(0, x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x1), 0, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, x1), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 7 }-> if_qs(0, low(n, x2), n', high(n'', x3)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 6 }-> if_qs(0, low(n, x2), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 6 }-> if_qs(0, low(n, 0), n', high(n'', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(n'', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), n', high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), n', high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(n, 0), 0, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(n, 0), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(n, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x'), 0, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, x'), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), n, high(0, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x''), 0, high(n, x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, x''), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 6 }-> if_qs(0, low(0, x1), n, high(n', x2)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 5 }-> if_qs(0, low(0, x1), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 5 }-> if_qs(0, low(0, 0), n, high(n', x1)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(n', 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), n, high(0, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), n, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 4 }-> if_qs(0, low(0, 0), 0, high(n, x'')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(n, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 3 }-> if_qs(0, low(0, 0), 0, high(0, x')) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 2 }-> if_qs(0, low(0, 0), 0, high(0, 0)) :|: z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']

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(57) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s152 :|: s152 >= 0, s152 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 42 + 8*x + 8*x1 }-> s153 :|: s153 >= 0, s153 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x2 + 2*x2*z + 11*z }-> s154 :|: s154 >= 0, s154 <= 1 + (1 + x2) + x, s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s155 :|: s155 >= 0, s155 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s156 :|: s156 >= 0, s156 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 34 + 3*s111 + 3*s114 }-> app(if_qs(1, low(s109, 0), s110, s193), 1 + z'' + if_qs(0, low(s112, x'), s113, s194)) :|: s193 >= 0, s193 <= 0, s194 >= 0, s194 <= 0, s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 34 + 3*s123 + 3*s126 + s126*x' + 4*x' }-> app(if_qs(1, low(s121, 0), s122, s197), 1 + z'' + if_qs(0, low(s124, 0), s125, s198)) :|: s197 >= 0, s197 <= 0, s198 >= 0, s198 <= x', s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 33 + 3*s141 + 3*s144 }-> app(if_qs(1, low(s139, 0), s140, s203), 1 + z'' + if_qs(0, low(s142, 0), s143, s204)) :|: s203 >= 0, s203 <= 0, s204 >= 0, s204 <= 0, s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 33 + 3*s147 + 3*s150 }-> app(if_qs(1, low(s145, 0), s146, s205), 1 + z'' + if_qs(1, low(s148, 0), s149, s206)) :|: s205 >= 0, s205 <= 0, s206 >= 0, s206 <= 0, s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s102 + s102*x'' + 3*s99 + 4*x'' }-> app(if_qs(1, low(s97, 0), s98, s189), 1 + z'' + if_qs(0, low(s100, x'), s101, s190)) :|: s189 >= 0, s189 <= 0, s190 >= 0, s190 <= x'', s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 37 + 3*s3 + s3*x'' + 3*s6 + s6*x2 + 4*x'' + 4*x2 }-> app(if_qs(0, low(s1, x'), s2, s157), 1 + z'' + if_qs(0, low(s4, x1), s5, s158)) :|: s157 >= 0, s157 <= x'', s158 >= 0, s158 <= x2, s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 34 + 3*s105 + 3*s108 }-> app(if_qs(0, low(s103, 0), s104, s191), 1 + z'' + if_qs(0, low(s106, x'), s107, s192)) :|: s191 >= 0, s191 <= 0, s192 >= 0, s192 <= 0, s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s117 + 3*s120 + s120*x' + 4*x' }-> app(if_qs(0, low(s115, 0), s116, s195), 1 + z'' + if_qs(0, low(s118, 0), s119, s196)) :|: s195 >= 0, s195 <= 0, s196 >= 0, s196 <= x', s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 33 + 3*s129 + 3*s132 }-> app(if_qs(0, low(s127, 0), s128, s199), 1 + z'' + if_qs(0, low(s130, 0), s131, s200)) :|: s199 >= 0, s199 <= 0, s200 >= 0, s200 <= 0, s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 36 + 3*s15 + s15*x'' + 3*s18 + s18*x1 + 4*x'' + 4*x1 }-> app(if_qs(0, low(s13, x'), s14, s161), 1 + z'' + if_qs(0, low(s16, 0), s17, s162)) :|: s161 >= 0, s161 <= x'', s162 >= 0, s162 <= x1, s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 33 + 3*s135 + 3*s138 }-> app(if_qs(0, low(s133, 0), s134, s201), 1 + z'' + if_qs(1, low(s136, 0), s137, s202)) :|: s201 >= 0, s201 <= 0, s202 >= 0, s202 <= 0, s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s21 + s21*x'' + 3*s24 + 4*x'' }-> app(if_qs(0, low(s19, x'), s20, s163), 1 + z'' + if_qs(0, low(s22, 0), s23, s164)) :|: s163 >= 0, s163 <= x'', s164 >= 0, s164 <= 0, s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 35 + 3*s27 + s27*x'' + 3*s30 + 4*x'' }-> app(if_qs(0, low(s25, x'), s26, s165), 1 + z'' + if_qs(1, low(s28, 0), s29, s166)) :|: s165 >= 0, s165 <= x'', s166 >= 0, s166 <= 0, s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 36 + 3*s33 + 3*s36 + s36*x1 + 4*x1 }-> app(if_qs(0, low(s31, x'), s32, s167), 1 + z'' + if_qs(0, low(s34, x''), s35, s168)) :|: s167 >= 0, s167 <= 0, s168 >= 0, s168 <= x1, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s39 + 3*s42 }-> app(if_qs(0, low(s37, x'), s38, s169), 1 + z'' + if_qs(0, low(s40, x''), s41, s170)) :|: s169 >= 0, s169 <= 0, s170 >= 0, s170 <= 0, s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 35 + 3*s45 + 3*s48 + s48*x'' + 4*x'' }-> app(if_qs(0, low(s43, x'), s44, s171), 1 + z'' + if_qs(0, low(s46, 0), s47, s172)) :|: s171 >= 0, s171 <= 0, s172 >= 0, s172 <= x'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 34 + 3*s51 + 3*s54 }-> app(if_qs(0, low(s49, x'), s50, s173), 1 + z'' + if_qs(0, low(s52, 0), s53, s174)) :|: s173 >= 0, s173 <= 0, s174 >= 0, s174 <= 0, s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s57 + 3*s60 }-> app(if_qs(0, low(s55, x'), s56, s175), 1 + z'' + if_qs(1, low(s58, 0), s59, s176)) :|: s175 >= 0, s175 <= 0, s176 >= 0, s176 <= 0, s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 36 + 3*s63 + s63*x' + 3*s66 + s66*x1 + 4*x' + 4*x1 }-> app(if_qs(0, low(s61, 0), s62, s177), 1 + z'' + if_qs(0, low(s64, x''), s65, s178)) :|: s177 >= 0, s177 <= x', s178 >= 0, s178 <= x1, s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s69 + s69*x' + 3*s72 + 4*x' }-> app(if_qs(0, low(s67, 0), s68, s179), 1 + z'' + if_qs(0, low(s70, x''), s71, s180)) :|: s179 >= 0, s179 <= x', s180 >= 0, s180 <= 0, s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 36 + 3*s12 + 3*s9 + s9*x'' + 4*x'' }-> app(if_qs(0, low(s7, x'), s8, s159), 1 + z'' + if_qs(0, low(s10, x1), s11, s160)) :|: s159 >= 0, s159 <= x'', s160 >= 0, s160 <= 0, s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s75 + s75*x' + 3*s78 + s78*x'' + 4*x' + 4*x'' }-> app(if_qs(0, low(s73, 0), s74, s181), 1 + z'' + if_qs(0, low(s76, 0), s77, s182)) :|: s181 >= 0, s181 <= x', s182 >= 0, s182 <= x'', s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 34 + 3*s81 + s81*x' + 3*s84 + 4*x' }-> app(if_qs(0, low(s79, 0), s80, s183), 1 + z'' + if_qs(0, low(s82, 0), s83, s184)) :|: s183 >= 0, s183 <= x', s184 >= 0, s184 <= 0, s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s87 + s87*x' + 3*s90 + 4*x' }-> app(if_qs(0, low(s85, 0), s86, s185), 1 + z'' + if_qs(1, low(s88, 0), s89, s186)) :|: s185 >= 0, s185 <= x', s186 >= 0, s186 <= 0, s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s93 + 3*s96 + s96*x'' + 4*x'' }-> app(if_qs(0, low(s91, 0), s92, s187), 1 + z'' + if_qs(0, low(s94, x'), s95, s188)) :|: s187 >= 0, s187 <= 0, s188 >= 0, s188 <= x'', s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 13 }-> if_qs(1, low(0, 0), 0, s239) :|: s239 >= 0, s239 <= 0, z >= 0, z = 0
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(n, x''), 0, s219) :|: s219 >= 0, s219 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(n, x''), 0, s220) :|: s220 >= 0, s220 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 17 + 4*x2 }-> if_qs(0, low(n, x1), n', s211) :|: s211 >= 0, s211 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 }-> if_qs(0, low(n, x1), n', s212) :|: s212 >= 0, s212 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 17 + 3*n' + n'*x2 + 4*x2 }-> if_qs(0, low(n, x1), 0, s215) :|: s215 >= 0, s215 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n' }-> if_qs(0, low(n, x1), 0, s216) :|: s216 >= 0, s216 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 18 + 3*n'' + n''*x3 + 4*x3 }-> if_qs(0, low(n, x2), n', s207) :|: s207 >= 0, s207 <= x3, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 17 + 3*n'' }-> if_qs(0, low(n, x2), n', s208) :|: s208 >= 0, s208 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 17 + 3*n'' + n''*x2 + 4*x2 }-> if_qs(0, low(n, 0), n', s209) :|: s209 >= 0, s209 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n'' }-> if_qs(0, low(n, 0), n', s210) :|: s210 >= 0, s210 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(n, 0), n', s213) :|: s213 >= 0, s213 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(n, 0), n', s214) :|: s214 >= 0, s214 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 16 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, low(n, 0), 0, s217) :|: s217 >= 0, s217 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n' }-> if_qs(0, low(n, 0), 0, s218) :|: s218 >= 0, s218 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(n, 0), 0, s221) :|: s221 >= 0, s221 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(n, 0), 0, s222) :|: s222 >= 0, s222 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(0, x'), 0, s235) :|: s235 >= 0, s235 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(0, x'), 0, s236) :|: s236 >= 0, s236 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(0, x''), n, s227) :|: s227 >= 0, s227 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(0, x''), n, s228) :|: s228 >= 0, s228 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 16 + 3*n + n*x1 + 4*x1 }-> if_qs(0, low(0, x''), 0, s231) :|: s231 >= 0, s231 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n }-> if_qs(0, low(0, x''), 0, s232) :|: s232 >= 0, s232 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 17 + 3*n' + n'*x2 + 4*x2 }-> if_qs(0, low(0, x1), n, s223) :|: s223 >= 0, s223 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n' }-> if_qs(0, low(0, x1), n, s224) :|: s224 >= 0, s224 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 16 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, low(0, 0), n, s225) :|: s225 >= 0, s225 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n' }-> if_qs(0, low(0, 0), n, s226) :|: s226 >= 0, s226 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(0, 0), n, s229) :|: s229 >= 0, s229 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(0, 0), n, s230) :|: s230 >= 0, s230 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 15 + 3*n + n*x'' + 4*x'' }-> if_qs(0, low(0, 0), 0, s233) :|: s233 >= 0, s233 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 + 3*n }-> if_qs(0, low(0, 0), 0, s234) :|: s234 >= 0, s234 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 14 + 4*x' }-> if_qs(0, low(0, 0), 0, s237) :|: s237 >= 0, s237 <= x', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 13 }-> if_qs(0, low(0, 0), 0, s238) :|: s238 >= 0, s238 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']

----------------------------------------

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: low
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

Computed SIZE bound using KoAT for: if_low
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z''

----------------------------------------

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s152 :|: s152 >= 0, s152 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 42 + 8*x + 8*x1 }-> s153 :|: s153 >= 0, s153 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x2 + 2*x2*z + 11*z }-> s154 :|: s154 >= 0, s154 <= 1 + (1 + x2) + x, s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s155 :|: s155 >= 0, s155 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s156 :|: s156 >= 0, s156 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 34 + 3*s111 + 3*s114 }-> app(if_qs(1, low(s109, 0), s110, s193), 1 + z'' + if_qs(0, low(s112, x'), s113, s194)) :|: s193 >= 0, s193 <= 0, s194 >= 0, s194 <= 0, s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 34 + 3*s123 + 3*s126 + s126*x' + 4*x' }-> app(if_qs(1, low(s121, 0), s122, s197), 1 + z'' + if_qs(0, low(s124, 0), s125, s198)) :|: s197 >= 0, s197 <= 0, s198 >= 0, s198 <= x', s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 33 + 3*s141 + 3*s144 }-> app(if_qs(1, low(s139, 0), s140, s203), 1 + z'' + if_qs(0, low(s142, 0), s143, s204)) :|: s203 >= 0, s203 <= 0, s204 >= 0, s204 <= 0, s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 33 + 3*s147 + 3*s150 }-> app(if_qs(1, low(s145, 0), s146, s205), 1 + z'' + if_qs(1, low(s148, 0), s149, s206)) :|: s205 >= 0, s205 <= 0, s206 >= 0, s206 <= 0, s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s102 + s102*x'' + 3*s99 + 4*x'' }-> app(if_qs(1, low(s97, 0), s98, s189), 1 + z'' + if_qs(0, low(s100, x'), s101, s190)) :|: s189 >= 0, s189 <= 0, s190 >= 0, s190 <= x'', s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 37 + 3*s3 + s3*x'' + 3*s6 + s6*x2 + 4*x'' + 4*x2 }-> app(if_qs(0, low(s1, x'), s2, s157), 1 + z'' + if_qs(0, low(s4, x1), s5, s158)) :|: s157 >= 0, s157 <= x'', s158 >= 0, s158 <= x2, s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 34 + 3*s105 + 3*s108 }-> app(if_qs(0, low(s103, 0), s104, s191), 1 + z'' + if_qs(0, low(s106, x'), s107, s192)) :|: s191 >= 0, s191 <= 0, s192 >= 0, s192 <= 0, s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s117 + 3*s120 + s120*x' + 4*x' }-> app(if_qs(0, low(s115, 0), s116, s195), 1 + z'' + if_qs(0, low(s118, 0), s119, s196)) :|: s195 >= 0, s195 <= 0, s196 >= 0, s196 <= x', s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 33 + 3*s129 + 3*s132 }-> app(if_qs(0, low(s127, 0), s128, s199), 1 + z'' + if_qs(0, low(s130, 0), s131, s200)) :|: s199 >= 0, s199 <= 0, s200 >= 0, s200 <= 0, s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 36 + 3*s15 + s15*x'' + 3*s18 + s18*x1 + 4*x'' + 4*x1 }-> app(if_qs(0, low(s13, x'), s14, s161), 1 + z'' + if_qs(0, low(s16, 0), s17, s162)) :|: s161 >= 0, s161 <= x'', s162 >= 0, s162 <= x1, s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 33 + 3*s135 + 3*s138 }-> app(if_qs(0, low(s133, 0), s134, s201), 1 + z'' + if_qs(1, low(s136, 0), s137, s202)) :|: s201 >= 0, s201 <= 0, s202 >= 0, s202 <= 0, s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s21 + s21*x'' + 3*s24 + 4*x'' }-> app(if_qs(0, low(s19, x'), s20, s163), 1 + z'' + if_qs(0, low(s22, 0), s23, s164)) :|: s163 >= 0, s163 <= x'', s164 >= 0, s164 <= 0, s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 35 + 3*s27 + s27*x'' + 3*s30 + 4*x'' }-> app(if_qs(0, low(s25, x'), s26, s165), 1 + z'' + if_qs(1, low(s28, 0), s29, s166)) :|: s165 >= 0, s165 <= x'', s166 >= 0, s166 <= 0, s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 36 + 3*s33 + 3*s36 + s36*x1 + 4*x1 }-> app(if_qs(0, low(s31, x'), s32, s167), 1 + z'' + if_qs(0, low(s34, x''), s35, s168)) :|: s167 >= 0, s167 <= 0, s168 >= 0, s168 <= x1, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s39 + 3*s42 }-> app(if_qs(0, low(s37, x'), s38, s169), 1 + z'' + if_qs(0, low(s40, x''), s41, s170)) :|: s169 >= 0, s169 <= 0, s170 >= 0, s170 <= 0, s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 35 + 3*s45 + 3*s48 + s48*x'' + 4*x'' }-> app(if_qs(0, low(s43, x'), s44, s171), 1 + z'' + if_qs(0, low(s46, 0), s47, s172)) :|: s171 >= 0, s171 <= 0, s172 >= 0, s172 <= x'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 34 + 3*s51 + 3*s54 }-> app(if_qs(0, low(s49, x'), s50, s173), 1 + z'' + if_qs(0, low(s52, 0), s53, s174)) :|: s173 >= 0, s173 <= 0, s174 >= 0, s174 <= 0, s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s57 + 3*s60 }-> app(if_qs(0, low(s55, x'), s56, s175), 1 + z'' + if_qs(1, low(s58, 0), s59, s176)) :|: s175 >= 0, s175 <= 0, s176 >= 0, s176 <= 0, s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 36 + 3*s63 + s63*x' + 3*s66 + s66*x1 + 4*x' + 4*x1 }-> app(if_qs(0, low(s61, 0), s62, s177), 1 + z'' + if_qs(0, low(s64, x''), s65, s178)) :|: s177 >= 0, s177 <= x', s178 >= 0, s178 <= x1, s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s69 + s69*x' + 3*s72 + 4*x' }-> app(if_qs(0, low(s67, 0), s68, s179), 1 + z'' + if_qs(0, low(s70, x''), s71, s180)) :|: s179 >= 0, s179 <= x', s180 >= 0, s180 <= 0, s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 36 + 3*s12 + 3*s9 + s9*x'' + 4*x'' }-> app(if_qs(0, low(s7, x'), s8, s159), 1 + z'' + if_qs(0, low(s10, x1), s11, s160)) :|: s159 >= 0, s159 <= x'', s160 >= 0, s160 <= 0, s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s75 + s75*x' + 3*s78 + s78*x'' + 4*x' + 4*x'' }-> app(if_qs(0, low(s73, 0), s74, s181), 1 + z'' + if_qs(0, low(s76, 0), s77, s182)) :|: s181 >= 0, s181 <= x', s182 >= 0, s182 <= x'', s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 34 + 3*s81 + s81*x' + 3*s84 + 4*x' }-> app(if_qs(0, low(s79, 0), s80, s183), 1 + z'' + if_qs(0, low(s82, 0), s83, s184)) :|: s183 >= 0, s183 <= x', s184 >= 0, s184 <= 0, s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s87 + s87*x' + 3*s90 + 4*x' }-> app(if_qs(0, low(s85, 0), s86, s185), 1 + z'' + if_qs(1, low(s88, 0), s89, s186)) :|: s185 >= 0, s185 <= x', s186 >= 0, s186 <= 0, s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s93 + 3*s96 + s96*x'' + 4*x'' }-> app(if_qs(0, low(s91, 0), s92, s187), 1 + z'' + if_qs(0, low(s94, x'), s95, s188)) :|: s187 >= 0, s187 <= 0, s188 >= 0, s188 <= x'', s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 13 }-> if_qs(1, low(0, 0), 0, s239) :|: s239 >= 0, s239 <= 0, z >= 0, z = 0
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(n, x''), 0, s219) :|: s219 >= 0, s219 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(n, x''), 0, s220) :|: s220 >= 0, s220 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 17 + 4*x2 }-> if_qs(0, low(n, x1), n', s211) :|: s211 >= 0, s211 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 }-> if_qs(0, low(n, x1), n', s212) :|: s212 >= 0, s212 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 17 + 3*n' + n'*x2 + 4*x2 }-> if_qs(0, low(n, x1), 0, s215) :|: s215 >= 0, s215 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n' }-> if_qs(0, low(n, x1), 0, s216) :|: s216 >= 0, s216 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 18 + 3*n'' + n''*x3 + 4*x3 }-> if_qs(0, low(n, x2), n', s207) :|: s207 >= 0, s207 <= x3, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 17 + 3*n'' }-> if_qs(0, low(n, x2), n', s208) :|: s208 >= 0, s208 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 17 + 3*n'' + n''*x2 + 4*x2 }-> if_qs(0, low(n, 0), n', s209) :|: s209 >= 0, s209 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n'' }-> if_qs(0, low(n, 0), n', s210) :|: s210 >= 0, s210 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(n, 0), n', s213) :|: s213 >= 0, s213 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(n, 0), n', s214) :|: s214 >= 0, s214 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 16 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, low(n, 0), 0, s217) :|: s217 >= 0, s217 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n' }-> if_qs(0, low(n, 0), 0, s218) :|: s218 >= 0, s218 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(n, 0), 0, s221) :|: s221 >= 0, s221 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(n, 0), 0, s222) :|: s222 >= 0, s222 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(0, x'), 0, s235) :|: s235 >= 0, s235 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(0, x'), 0, s236) :|: s236 >= 0, s236 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(0, x''), n, s227) :|: s227 >= 0, s227 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(0, x''), n, s228) :|: s228 >= 0, s228 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 16 + 3*n + n*x1 + 4*x1 }-> if_qs(0, low(0, x''), 0, s231) :|: s231 >= 0, s231 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n }-> if_qs(0, low(0, x''), 0, s232) :|: s232 >= 0, s232 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 17 + 3*n' + n'*x2 + 4*x2 }-> if_qs(0, low(0, x1), n, s223) :|: s223 >= 0, s223 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n' }-> if_qs(0, low(0, x1), n, s224) :|: s224 >= 0, s224 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 16 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, low(0, 0), n, s225) :|: s225 >= 0, s225 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n' }-> if_qs(0, low(0, 0), n, s226) :|: s226 >= 0, s226 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(0, 0), n, s229) :|: s229 >= 0, s229 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(0, 0), n, s230) :|: s230 >= 0, s230 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 15 + 3*n + n*x'' + 4*x'' }-> if_qs(0, low(0, 0), 0, s233) :|: s233 >= 0, s233 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 + 3*n }-> if_qs(0, low(0, 0), 0, s234) :|: s234 >= 0, s234 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 14 + 4*x' }-> if_qs(0, low(0, 0), 0, s237) :|: s237 >= 0, s237 <= x', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 13 }-> if_qs(0, low(0, 0), 0, s238) :|: s238 >= 0, s238 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {low,if_low}, {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  low: runtime: ?, size: O(n^1) [z']
  if_low: runtime: ?, size: O(n^1) [z'']

----------------------------------------

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: low
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 11 + 3*z + z*z' + 4*z'

Computed RUNTIME bound using KoAT for: if_low
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z'' + 8*z''

----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s152 :|: s152 >= 0, s152 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 42 + 8*x + 8*x1 }-> s153 :|: s153 >= 0, s153 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x2 + 2*x2*z + 11*z }-> s154 :|: s154 >= 0, s154 <= 1 + (1 + x2) + x, s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s155 :|: s155 >= 0, s155 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s156 :|: s156 >= 0, s156 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(z', x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(z', x) :|: z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 34 + 3*s111 + 3*s114 }-> app(if_qs(1, low(s109, 0), s110, s193), 1 + z'' + if_qs(0, low(s112, x'), s113, s194)) :|: s193 >= 0, s193 <= 0, s194 >= 0, s194 <= 0, s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 34 + 3*s123 + 3*s126 + s126*x' + 4*x' }-> app(if_qs(1, low(s121, 0), s122, s197), 1 + z'' + if_qs(0, low(s124, 0), s125, s198)) :|: s197 >= 0, s197 <= 0, s198 >= 0, s198 <= x', s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 33 + 3*s141 + 3*s144 }-> app(if_qs(1, low(s139, 0), s140, s203), 1 + z'' + if_qs(0, low(s142, 0), s143, s204)) :|: s203 >= 0, s203 <= 0, s204 >= 0, s204 <= 0, s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 33 + 3*s147 + 3*s150 }-> app(if_qs(1, low(s145, 0), s146, s205), 1 + z'' + if_qs(1, low(s148, 0), s149, s206)) :|: s205 >= 0, s205 <= 0, s206 >= 0, s206 <= 0, s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s102 + s102*x'' + 3*s99 + 4*x'' }-> app(if_qs(1, low(s97, 0), s98, s189), 1 + z'' + if_qs(0, low(s100, x'), s101, s190)) :|: s189 >= 0, s189 <= 0, s190 >= 0, s190 <= x'', s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 37 + 3*s3 + s3*x'' + 3*s6 + s6*x2 + 4*x'' + 4*x2 }-> app(if_qs(0, low(s1, x'), s2, s157), 1 + z'' + if_qs(0, low(s4, x1), s5, s158)) :|: s157 >= 0, s157 <= x'', s158 >= 0, s158 <= x2, s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 34 + 3*s105 + 3*s108 }-> app(if_qs(0, low(s103, 0), s104, s191), 1 + z'' + if_qs(0, low(s106, x'), s107, s192)) :|: s191 >= 0, s191 <= 0, s192 >= 0, s192 <= 0, s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s117 + 3*s120 + s120*x' + 4*x' }-> app(if_qs(0, low(s115, 0), s116, s195), 1 + z'' + if_qs(0, low(s118, 0), s119, s196)) :|: s195 >= 0, s195 <= 0, s196 >= 0, s196 <= x', s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 33 + 3*s129 + 3*s132 }-> app(if_qs(0, low(s127, 0), s128, s199), 1 + z'' + if_qs(0, low(s130, 0), s131, s200)) :|: s199 >= 0, s199 <= 0, s200 >= 0, s200 <= 0, s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 36 + 3*s15 + s15*x'' + 3*s18 + s18*x1 + 4*x'' + 4*x1 }-> app(if_qs(0, low(s13, x'), s14, s161), 1 + z'' + if_qs(0, low(s16, 0), s17, s162)) :|: s161 >= 0, s161 <= x'', s162 >= 0, s162 <= x1, s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 33 + 3*s135 + 3*s138 }-> app(if_qs(0, low(s133, 0), s134, s201), 1 + z'' + if_qs(1, low(s136, 0), s137, s202)) :|: s201 >= 0, s201 <= 0, s202 >= 0, s202 <= 0, s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s21 + s21*x'' + 3*s24 + 4*x'' }-> app(if_qs(0, low(s19, x'), s20, s163), 1 + z'' + if_qs(0, low(s22, 0), s23, s164)) :|: s163 >= 0, s163 <= x'', s164 >= 0, s164 <= 0, s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 35 + 3*s27 + s27*x'' + 3*s30 + 4*x'' }-> app(if_qs(0, low(s25, x'), s26, s165), 1 + z'' + if_qs(1, low(s28, 0), s29, s166)) :|: s165 >= 0, s165 <= x'', s166 >= 0, s166 <= 0, s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 36 + 3*s33 + 3*s36 + s36*x1 + 4*x1 }-> app(if_qs(0, low(s31, x'), s32, s167), 1 + z'' + if_qs(0, low(s34, x''), s35, s168)) :|: s167 >= 0, s167 <= 0, s168 >= 0, s168 <= x1, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s39 + 3*s42 }-> app(if_qs(0, low(s37, x'), s38, s169), 1 + z'' + if_qs(0, low(s40, x''), s41, s170)) :|: s169 >= 0, s169 <= 0, s170 >= 0, s170 <= 0, s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 35 + 3*s45 + 3*s48 + s48*x'' + 4*x'' }-> app(if_qs(0, low(s43, x'), s44, s171), 1 + z'' + if_qs(0, low(s46, 0), s47, s172)) :|: s171 >= 0, s171 <= 0, s172 >= 0, s172 <= x'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 34 + 3*s51 + 3*s54 }-> app(if_qs(0, low(s49, x'), s50, s173), 1 + z'' + if_qs(0, low(s52, 0), s53, s174)) :|: s173 >= 0, s173 <= 0, s174 >= 0, s174 <= 0, s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s57 + 3*s60 }-> app(if_qs(0, low(s55, x'), s56, s175), 1 + z'' + if_qs(1, low(s58, 0), s59, s176)) :|: s175 >= 0, s175 <= 0, s176 >= 0, s176 <= 0, s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 36 + 3*s63 + s63*x' + 3*s66 + s66*x1 + 4*x' + 4*x1 }-> app(if_qs(0, low(s61, 0), s62, s177), 1 + z'' + if_qs(0, low(s64, x''), s65, s178)) :|: s177 >= 0, s177 <= x', s178 >= 0, s178 <= x1, s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s69 + s69*x' + 3*s72 + 4*x' }-> app(if_qs(0, low(s67, 0), s68, s179), 1 + z'' + if_qs(0, low(s70, x''), s71, s180)) :|: s179 >= 0, s179 <= x', s180 >= 0, s180 <= 0, s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 36 + 3*s12 + 3*s9 + s9*x'' + 4*x'' }-> app(if_qs(0, low(s7, x'), s8, s159), 1 + z'' + if_qs(0, low(s10, x1), s11, s160)) :|: s159 >= 0, s159 <= x'', s160 >= 0, s160 <= 0, s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 35 + 3*s75 + s75*x' + 3*s78 + s78*x'' + 4*x' + 4*x'' }-> app(if_qs(0, low(s73, 0), s74, s181), 1 + z'' + if_qs(0, low(s76, 0), s77, s182)) :|: s181 >= 0, s181 <= x', s182 >= 0, s182 <= x'', s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 34 + 3*s81 + s81*x' + 3*s84 + 4*x' }-> app(if_qs(0, low(s79, 0), s80, s183), 1 + z'' + if_qs(0, low(s82, 0), s83, s184)) :|: s183 >= 0, s183 <= x', s184 >= 0, s184 <= 0, s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 34 + 3*s87 + s87*x' + 3*s90 + 4*x' }-> app(if_qs(0, low(s85, 0), s86, s185), 1 + z'' + if_qs(1, low(s88, 0), s89, s186)) :|: s185 >= 0, s185 <= x', s186 >= 0, s186 <= 0, s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 35 + 3*s93 + 3*s96 + s96*x'' + 4*x'' }-> app(if_qs(0, low(s91, 0), s92, s187), 1 + z'' + if_qs(0, low(s94, x'), s95, s188)) :|: s187 >= 0, s187 <= 0, s188 >= 0, s188 <= x'', s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 3 + z }-> if_low(s', 1 + (z - 1), 1 + (1 + x'') + x) :|: s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 2 }-> if_low(1, z, 1 + 0 + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   low(z, z') -{ 2 }-> if_low(0, 0, 1 + (1 + x') + x) :|: x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 13 }-> if_qs(1, low(0, 0), 0, s239) :|: s239 >= 0, s239 <= 0, z >= 0, z = 0
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(n, x''), 0, s219) :|: s219 >= 0, s219 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(n, x''), 0, s220) :|: s220 >= 0, s220 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 17 + 4*x2 }-> if_qs(0, low(n, x1), n', s211) :|: s211 >= 0, s211 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 }-> if_qs(0, low(n, x1), n', s212) :|: s212 >= 0, s212 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 17 + 3*n' + n'*x2 + 4*x2 }-> if_qs(0, low(n, x1), 0, s215) :|: s215 >= 0, s215 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n' }-> if_qs(0, low(n, x1), 0, s216) :|: s216 >= 0, s216 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 18 + 3*n'' + n''*x3 + 4*x3 }-> if_qs(0, low(n, x2), n', s207) :|: s207 >= 0, s207 <= x3, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 17 + 3*n'' }-> if_qs(0, low(n, x2), n', s208) :|: s208 >= 0, s208 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 17 + 3*n'' + n''*x2 + 4*x2 }-> if_qs(0, low(n, 0), n', s209) :|: s209 >= 0, s209 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n'' }-> if_qs(0, low(n, 0), n', s210) :|: s210 >= 0, s210 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(n, 0), n', s213) :|: s213 >= 0, s213 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(n, 0), n', s214) :|: s214 >= 0, s214 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 16 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, low(n, 0), 0, s217) :|: s217 >= 0, s217 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n' }-> if_qs(0, low(n, 0), 0, s218) :|: s218 >= 0, s218 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(n, 0), 0, s221) :|: s221 >= 0, s221 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(n, 0), 0, s222) :|: s222 >= 0, s222 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(0, x'), 0, s235) :|: s235 >= 0, s235 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(0, x'), 0, s236) :|: s236 >= 0, s236 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 16 + 4*x1 }-> if_qs(0, low(0, x''), n, s227) :|: s227 >= 0, s227 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 }-> if_qs(0, low(0, x''), n, s228) :|: s228 >= 0, s228 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 16 + 3*n + n*x1 + 4*x1 }-> if_qs(0, low(0, x''), 0, s231) :|: s231 >= 0, s231 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n }-> if_qs(0, low(0, x''), 0, s232) :|: s232 >= 0, s232 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 17 + 3*n' + n'*x2 + 4*x2 }-> if_qs(0, low(0, x1), n, s223) :|: s223 >= 0, s223 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 16 + 3*n' }-> if_qs(0, low(0, x1), n, s224) :|: s224 >= 0, s224 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 16 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, low(0, 0), n, s225) :|: s225 >= 0, s225 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 15 + 3*n' }-> if_qs(0, low(0, 0), n, s226) :|: s226 >= 0, s226 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 15 + 4*x'' }-> if_qs(0, low(0, 0), n, s229) :|: s229 >= 0, s229 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 }-> if_qs(0, low(0, 0), n, s230) :|: s230 >= 0, s230 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 15 + 3*n + n*x'' + 4*x'' }-> if_qs(0, low(0, 0), 0, s233) :|: s233 >= 0, s233 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 14 + 3*n }-> if_qs(0, low(0, 0), 0, s234) :|: s234 >= 0, s234 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 14 + 4*x' }-> if_qs(0, low(0, 0), 0, s237) :|: s237 >= 0, s237 <= x', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 13 }-> if_qs(0, low(0, 0), 0, s238) :|: s238 >= 0, s238 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']

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(63) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s152 :|: s152 >= 0, s152 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 42 + 8*x + 8*x1 }-> s153 :|: s153 >= 0, s153 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x2 + 2*x2*z + 11*z }-> s154 :|: s154 >= 0, s154 <= 1 + (1 + x2) + x, s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s155 :|: s155 >= 0, s155 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s156 :|: s156 >= 0, s156 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s244 :|: s244 >= 0, s244 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s243 :|: s243 >= 0, s243 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 57 + 3*s100 + s100*x' + 3*s102 + s102*x'' + 3*s97 + 3*s99 + 4*x' + 4*x'' }-> app(if_qs(1, s277, s98, s189), 1 + z'' + if_qs(0, s278, s101, s190)) :|: s277 >= 0, s277 <= 0, s278 >= 0, s278 <= x', s189 >= 0, s189 <= 0, s190 >= 0, s190 <= x'', s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s109 + 3*s111 + 3*s112 + s112*x' + 3*s114 + 4*x' }-> app(if_qs(1, s281, s110, s193), 1 + z'' + if_qs(0, s282, s113, s194)) :|: s281 >= 0, s281 <= 0, s282 >= 0, s282 <= x', s193 >= 0, s193 <= 0, s194 >= 0, s194 <= 0, s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 56 + 3*s121 + 3*s123 + 3*s124 + 3*s126 + s126*x' + 4*x' }-> app(if_qs(1, s285, s122, s197), 1 + z'' + if_qs(0, s286, s125, s198)) :|: s285 >= 0, s285 <= 0, s286 >= 0, s286 <= 0, s197 >= 0, s197 <= 0, s198 >= 0, s198 <= x', s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 55 + 3*s139 + 3*s141 + 3*s142 + 3*s144 }-> app(if_qs(1, s291, s140, s203), 1 + z'' + if_qs(0, s292, s143, s204)) :|: s291 >= 0, s291 <= 0, s292 >= 0, s292 <= 0, s203 >= 0, s203 <= 0, s204 >= 0, s204 <= 0, s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 55 + 3*s145 + 3*s147 + 3*s148 + 3*s150 }-> app(if_qs(1, s293, s146, s205), 1 + z'' + if_qs(1, s294, s149, s206)) :|: s293 >= 0, s293 <= 0, s294 >= 0, s294 <= 0, s205 >= 0, s205 <= 0, s206 >= 0, s206 <= 0, s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 59 + 3*s1 + s1*x' + 3*s3 + s3*x'' + 3*s4 + s4*x1 + 3*s6 + s6*x2 + 4*x' + 4*x'' + 4*x1 + 4*x2 }-> app(if_qs(0, s245, s2, s157), 1 + z'' + if_qs(0, s246, s5, s158)) :|: s245 >= 0, s245 <= x', s246 >= 0, s246 <= x1, s157 >= 0, s157 <= x'', s158 >= 0, s158 <= x2, s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 58 + 3*s10 + s10*x1 + 3*s12 + 3*s7 + s7*x' + 3*s9 + s9*x'' + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s247, s8, s159), 1 + z'' + if_qs(0, s248, s11, s160)) :|: s247 >= 0, s247 <= x', s248 >= 0, s248 <= x1, s159 >= 0, s159 <= x'', s160 >= 0, s160 <= 0, s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 58 + 3*s13 + s13*x' + 3*s15 + s15*x'' + 3*s16 + 3*s18 + s18*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s249, s14, s161), 1 + z'' + if_qs(0, s250, s17, s162)) :|: s249 >= 0, s249 <= x', s250 >= 0, s250 <= 0, s161 >= 0, s161 <= x'', s162 >= 0, s162 <= x1, s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s19 + s19*x' + 3*s21 + s21*x'' + 3*s22 + 3*s24 + 4*x' + 4*x'' }-> app(if_qs(0, s251, s20, s163), 1 + z'' + if_qs(0, s252, s23, s164)) :|: s251 >= 0, s251 <= x', s252 >= 0, s252 <= 0, s163 >= 0, s163 <= x'', s164 >= 0, s164 <= 0, s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s25 + s25*x' + 3*s27 + s27*x'' + 3*s28 + 3*s30 + 4*x' + 4*x'' }-> app(if_qs(0, s253, s26, s165), 1 + z'' + if_qs(1, s254, s29, s166)) :|: s253 >= 0, s253 <= x', s254 >= 0, s254 <= 0, s165 >= 0, s165 <= x'', s166 >= 0, s166 <= 0, s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 58 + 3*s31 + s31*x' + 3*s33 + 3*s34 + s34*x'' + 3*s36 + s36*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s255, s32, s167), 1 + z'' + if_qs(0, s256, s35, s168)) :|: s255 >= 0, s255 <= x', s256 >= 0, s256 <= x'', s167 >= 0, s167 <= 0, s168 >= 0, s168 <= x1, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s37 + s37*x' + 3*s39 + 3*s40 + s40*x'' + 3*s42 + 4*x' + 4*x'' }-> app(if_qs(0, s257, s38, s169), 1 + z'' + if_qs(0, s258, s41, s170)) :|: s257 >= 0, s257 <= x', s258 >= 0, s258 <= x'', s169 >= 0, s169 <= 0, s170 >= 0, s170 <= 0, s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s43 + s43*x' + 3*s45 + 3*s46 + 3*s48 + s48*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s259, s44, s171), 1 + z'' + if_qs(0, s260, s47, s172)) :|: s259 >= 0, s259 <= x', s260 >= 0, s260 <= 0, s171 >= 0, s171 <= 0, s172 >= 0, s172 <= x'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s49 + s49*x' + 3*s51 + 3*s52 + 3*s54 + 4*x' }-> app(if_qs(0, s261, s50, s173), 1 + z'' + if_qs(0, s262, s53, s174)) :|: s261 >= 0, s261 <= x', s262 >= 0, s262 <= 0, s173 >= 0, s173 <= 0, s174 >= 0, s174 <= 0, s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s55 + s55*x' + 3*s57 + 3*s58 + 3*s60 + 4*x' }-> app(if_qs(0, s263, s56, s175), 1 + z'' + if_qs(1, s264, s59, s176)) :|: s263 >= 0, s263 <= x', s264 >= 0, s264 <= 0, s175 >= 0, s175 <= 0, s176 >= 0, s176 <= 0, s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 58 + 3*s61 + 3*s63 + s63*x' + 3*s64 + s64*x'' + 3*s66 + s66*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s265, s62, s177), 1 + z'' + if_qs(0, s266, s65, s178)) :|: s265 >= 0, s265 <= 0, s266 >= 0, s266 <= x'', s177 >= 0, s177 <= x', s178 >= 0, s178 <= x1, s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s67 + 3*s69 + s69*x' + 3*s70 + s70*x'' + 3*s72 + 4*x' + 4*x'' }-> app(if_qs(0, s267, s68, s179), 1 + z'' + if_qs(0, s268, s71, s180)) :|: s267 >= 0, s267 <= 0, s268 >= 0, s268 <= x'', s179 >= 0, s179 <= x', s180 >= 0, s180 <= 0, s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s73 + 3*s75 + s75*x' + 3*s76 + 3*s78 + s78*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s269, s74, s181), 1 + z'' + if_qs(0, s270, s77, s182)) :|: s269 >= 0, s269 <= 0, s270 >= 0, s270 <= 0, s181 >= 0, s181 <= x', s182 >= 0, s182 <= x'', s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s79 + 3*s81 + s81*x' + 3*s82 + 3*s84 + 4*x' }-> app(if_qs(0, s271, s80, s183), 1 + z'' + if_qs(0, s272, s83, s184)) :|: s271 >= 0, s271 <= 0, s272 >= 0, s272 <= 0, s183 >= 0, s183 <= x', s184 >= 0, s184 <= 0, s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s85 + 3*s87 + s87*x' + 3*s88 + 3*s90 + 4*x' }-> app(if_qs(0, s273, s86, s185), 1 + z'' + if_qs(1, s274, s89, s186)) :|: s273 >= 0, s273 <= 0, s274 >= 0, s274 <= 0, s185 >= 0, s185 <= x', s186 >= 0, s186 <= 0, s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 57 + 3*s91 + 3*s93 + 3*s94 + s94*x' + 3*s96 + s96*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s275, s92, s187), 1 + z'' + if_qs(0, s276, s95, s188)) :|: s275 >= 0, s275 <= 0, s276 >= 0, s276 <= x', s187 >= 0, s187 <= 0, s188 >= 0, s188 <= x'', s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s103 + 3*s105 + 3*s106 + s106*x' + 3*s108 + 4*x' }-> app(if_qs(0, s279, s104, s191), 1 + z'' + if_qs(0, s280, s107, s192)) :|: s279 >= 0, s279 <= 0, s280 >= 0, s280 <= x', s191 >= 0, s191 <= 0, s192 >= 0, s192 <= 0, s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s115 + 3*s117 + 3*s118 + 3*s120 + s120*x' + 4*x' }-> app(if_qs(0, s283, s116, s195), 1 + z'' + if_qs(0, s284, s119, s196)) :|: s283 >= 0, s283 <= 0, s284 >= 0, s284 <= 0, s195 >= 0, s195 <= 0, s196 >= 0, s196 <= x', s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 55 + 3*s127 + 3*s129 + 3*s130 + 3*s132 }-> app(if_qs(0, s287, s128, s199), 1 + z'' + if_qs(0, s288, s131, s200)) :|: s287 >= 0, s287 <= 0, s288 >= 0, s288 <= 0, s199 >= 0, s199 <= 0, s200 >= 0, s200 <= 0, s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 55 + 3*s133 + 3*s135 + 3*s136 + 3*s138 }-> app(if_qs(0, s289, s134, s201), 1 + z'' + if_qs(1, s290, s137, s202)) :|: s289 >= 0, s289 <= 0, s290 >= 0, s290 <= 0, s201 >= 0, s201 <= 0, s202 >= 0, s202 <= 0, s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s240 :|: s240 >= 0, s240 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   low(z, z') -{ 42 + 8*x + 8*x' }-> s241 :|: s241 >= 0, s241 <= 1 + (1 + x') + x, x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 43 + 8*x + 2*x*z + 8*x'' + 2*x''*z + 11*z }-> s242 :|: s242 >= 0, s242 <= 1 + (1 + x'') + x, s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 24 }-> if_qs(1, s327, 0, s239) :|: s327 >= 0, s327 <= 0, s239 >= 0, s239 <= 0, z >= 0, z = 0
   quicksort(z) -{ 29 + 3*n + n*x2 + 3*n'' + n''*x3 + 4*x2 + 4*x3 }-> if_qs(0, s295, n', s207) :|: s295 >= 0, s295 <= x2, s207 >= 0, s207 <= x3, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 28 + 3*n + n*x2 + 3*n'' + 4*x2 }-> if_qs(0, s296, n', s208) :|: s296 >= 0, s296 <= x2, s208 >= 0, s208 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 28 + 3*n + 3*n'' + n''*x2 + 4*x2 }-> if_qs(0, s297, n', s209) :|: s297 >= 0, s297 <= 0, s209 >= 0, s209 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + 3*n'' }-> if_qs(0, s298, n', s210) :|: s298 >= 0, s298 <= 0, s210 >= 0, s210 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 28 + 3*n + n*x1 + 4*x1 + 4*x2 }-> if_qs(0, s299, n', s211) :|: s299 >= 0, s299 <= x1, s211 >= 0, s211 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + n*x1 + 4*x1 }-> if_qs(0, s300, n', s212) :|: s300 >= 0, s300 <= x1, s212 >= 0, s212 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n + 4*x1 }-> if_qs(0, s301, n', s213) :|: s301 >= 0, s301 <= 0, s213 >= 0, s213 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n }-> if_qs(0, s302, n', s214) :|: s302 >= 0, s302 <= 0, s214 >= 0, s214 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 28 + 3*n + n*x1 + 3*n' + n'*x2 + 4*x1 + 4*x2 }-> if_qs(0, s303, 0, s215) :|: s303 >= 0, s303 <= x1, s215 >= 0, s215 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + n*x1 + 3*n' + 4*x1 }-> if_qs(0, s304, 0, s216) :|: s304 >= 0, s304 <= x1, s216 >= 0, s216 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, s305, 0, s217) :|: s305 >= 0, s305 <= 0, s217 >= 0, s217 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + 3*n' }-> if_qs(0, s306, 0, s218) :|: s306 >= 0, s306 <= 0, s218 >= 0, s218 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 27 + 3*n + n*x'' + 4*x'' + 4*x1 }-> if_qs(0, s307, 0, s219) :|: s307 >= 0, s307 <= x'', s219 >= 0, s219 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + n*x'' + 4*x'' }-> if_qs(0, s308, 0, s220) :|: s308 >= 0, s308 <= x'', s220 >= 0, s220 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 3*n + 4*x'' }-> if_qs(0, s309, 0, s221) :|: s309 >= 0, s309 <= 0, s221 >= 0, s221 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 3*n }-> if_qs(0, s310, 0, s222) :|: s310 >= 0, s310 <= 0, s222 >= 0, s222 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 28 + 3*n' + n'*x2 + 4*x1 + 4*x2 }-> if_qs(0, s311, n, s223) :|: s311 >= 0, s311 <= x1, s223 >= 0, s223 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n' + 4*x1 }-> if_qs(0, s312, n, s224) :|: s312 >= 0, s312 <= x1, s224 >= 0, s224 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, s313, n, s225) :|: s313 >= 0, s313 <= 0, s225 >= 0, s225 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n' }-> if_qs(0, s314, n, s226) :|: s314 >= 0, s314 <= 0, s226 >= 0, s226 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 27 + 4*x'' + 4*x1 }-> if_qs(0, s315, n, s227) :|: s315 >= 0, s315 <= x'', s227 >= 0, s227 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 4*x'' }-> if_qs(0, s316, n, s228) :|: s316 >= 0, s316 <= x'', s228 >= 0, s228 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 4*x'' }-> if_qs(0, s317, n, s229) :|: s317 >= 0, s317 <= 0, s229 >= 0, s229 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 }-> if_qs(0, s318, n, s230) :|: s318 >= 0, s318 <= 0, s230 >= 0, s230 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 27 + 3*n + n*x1 + 4*x'' + 4*x1 }-> if_qs(0, s319, 0, s231) :|: s319 >= 0, s319 <= x'', s231 >= 0, s231 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + 4*x'' }-> if_qs(0, s320, 0, s232) :|: s320 >= 0, s320 <= x'', s232 >= 0, s232 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 3*n + n*x'' + 4*x'' }-> if_qs(0, s321, 0, s233) :|: s321 >= 0, s321 <= 0, s233 >= 0, s233 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 3*n }-> if_qs(0, s322, 0, s234) :|: s322 >= 0, s322 <= 0, s234 >= 0, s234 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 26 + 4*x' + 4*x'' }-> if_qs(0, s323, 0, s235) :|: s323 >= 0, s323 <= x', s235 >= 0, s235 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 4*x' }-> if_qs(0, s324, 0, s236) :|: s324 >= 0, s324 <= x', s236 >= 0, s236 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 25 + 4*x' }-> if_qs(0, s325, 0, s237) :|: s325 >= 0, s325 <= 0, s237 >= 0, s237 <= x', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 24 }-> if_qs(0, s326, 0, s238) :|: s326 >= 0, s326 <= 0, s238 >= 0, s238 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']

----------------------------------------

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if_qs
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s152 :|: s152 >= 0, s152 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 42 + 8*x + 8*x1 }-> s153 :|: s153 >= 0, s153 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x2 + 2*x2*z + 11*z }-> s154 :|: s154 >= 0, s154 <= 1 + (1 + x2) + x, s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s155 :|: s155 >= 0, s155 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s156 :|: s156 >= 0, s156 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s244 :|: s244 >= 0, s244 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s243 :|: s243 >= 0, s243 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 57 + 3*s100 + s100*x' + 3*s102 + s102*x'' + 3*s97 + 3*s99 + 4*x' + 4*x'' }-> app(if_qs(1, s277, s98, s189), 1 + z'' + if_qs(0, s278, s101, s190)) :|: s277 >= 0, s277 <= 0, s278 >= 0, s278 <= x', s189 >= 0, s189 <= 0, s190 >= 0, s190 <= x'', s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s109 + 3*s111 + 3*s112 + s112*x' + 3*s114 + 4*x' }-> app(if_qs(1, s281, s110, s193), 1 + z'' + if_qs(0, s282, s113, s194)) :|: s281 >= 0, s281 <= 0, s282 >= 0, s282 <= x', s193 >= 0, s193 <= 0, s194 >= 0, s194 <= 0, s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 56 + 3*s121 + 3*s123 + 3*s124 + 3*s126 + s126*x' + 4*x' }-> app(if_qs(1, s285, s122, s197), 1 + z'' + if_qs(0, s286, s125, s198)) :|: s285 >= 0, s285 <= 0, s286 >= 0, s286 <= 0, s197 >= 0, s197 <= 0, s198 >= 0, s198 <= x', s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 55 + 3*s139 + 3*s141 + 3*s142 + 3*s144 }-> app(if_qs(1, s291, s140, s203), 1 + z'' + if_qs(0, s292, s143, s204)) :|: s291 >= 0, s291 <= 0, s292 >= 0, s292 <= 0, s203 >= 0, s203 <= 0, s204 >= 0, s204 <= 0, s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 55 + 3*s145 + 3*s147 + 3*s148 + 3*s150 }-> app(if_qs(1, s293, s146, s205), 1 + z'' + if_qs(1, s294, s149, s206)) :|: s293 >= 0, s293 <= 0, s294 >= 0, s294 <= 0, s205 >= 0, s205 <= 0, s206 >= 0, s206 <= 0, s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 59 + 3*s1 + s1*x' + 3*s3 + s3*x'' + 3*s4 + s4*x1 + 3*s6 + s6*x2 + 4*x' + 4*x'' + 4*x1 + 4*x2 }-> app(if_qs(0, s245, s2, s157), 1 + z'' + if_qs(0, s246, s5, s158)) :|: s245 >= 0, s245 <= x', s246 >= 0, s246 <= x1, s157 >= 0, s157 <= x'', s158 >= 0, s158 <= x2, s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 58 + 3*s10 + s10*x1 + 3*s12 + 3*s7 + s7*x' + 3*s9 + s9*x'' + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s247, s8, s159), 1 + z'' + if_qs(0, s248, s11, s160)) :|: s247 >= 0, s247 <= x', s248 >= 0, s248 <= x1, s159 >= 0, s159 <= x'', s160 >= 0, s160 <= 0, s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 58 + 3*s13 + s13*x' + 3*s15 + s15*x'' + 3*s16 + 3*s18 + s18*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s249, s14, s161), 1 + z'' + if_qs(0, s250, s17, s162)) :|: s249 >= 0, s249 <= x', s250 >= 0, s250 <= 0, s161 >= 0, s161 <= x'', s162 >= 0, s162 <= x1, s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s19 + s19*x' + 3*s21 + s21*x'' + 3*s22 + 3*s24 + 4*x' + 4*x'' }-> app(if_qs(0, s251, s20, s163), 1 + z'' + if_qs(0, s252, s23, s164)) :|: s251 >= 0, s251 <= x', s252 >= 0, s252 <= 0, s163 >= 0, s163 <= x'', s164 >= 0, s164 <= 0, s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s25 + s25*x' + 3*s27 + s27*x'' + 3*s28 + 3*s30 + 4*x' + 4*x'' }-> app(if_qs(0, s253, s26, s165), 1 + z'' + if_qs(1, s254, s29, s166)) :|: s253 >= 0, s253 <= x', s254 >= 0, s254 <= 0, s165 >= 0, s165 <= x'', s166 >= 0, s166 <= 0, s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 58 + 3*s31 + s31*x' + 3*s33 + 3*s34 + s34*x'' + 3*s36 + s36*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s255, s32, s167), 1 + z'' + if_qs(0, s256, s35, s168)) :|: s255 >= 0, s255 <= x', s256 >= 0, s256 <= x'', s167 >= 0, s167 <= 0, s168 >= 0, s168 <= x1, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s37 + s37*x' + 3*s39 + 3*s40 + s40*x'' + 3*s42 + 4*x' + 4*x'' }-> app(if_qs(0, s257, s38, s169), 1 + z'' + if_qs(0, s258, s41, s170)) :|: s257 >= 0, s257 <= x', s258 >= 0, s258 <= x'', s169 >= 0, s169 <= 0, s170 >= 0, s170 <= 0, s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s43 + s43*x' + 3*s45 + 3*s46 + 3*s48 + s48*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s259, s44, s171), 1 + z'' + if_qs(0, s260, s47, s172)) :|: s259 >= 0, s259 <= x', s260 >= 0, s260 <= 0, s171 >= 0, s171 <= 0, s172 >= 0, s172 <= x'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s49 + s49*x' + 3*s51 + 3*s52 + 3*s54 + 4*x' }-> app(if_qs(0, s261, s50, s173), 1 + z'' + if_qs(0, s262, s53, s174)) :|: s261 >= 0, s261 <= x', s262 >= 0, s262 <= 0, s173 >= 0, s173 <= 0, s174 >= 0, s174 <= 0, s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s55 + s55*x' + 3*s57 + 3*s58 + 3*s60 + 4*x' }-> app(if_qs(0, s263, s56, s175), 1 + z'' + if_qs(1, s264, s59, s176)) :|: s263 >= 0, s263 <= x', s264 >= 0, s264 <= 0, s175 >= 0, s175 <= 0, s176 >= 0, s176 <= 0, s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 58 + 3*s61 + 3*s63 + s63*x' + 3*s64 + s64*x'' + 3*s66 + s66*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s265, s62, s177), 1 + z'' + if_qs(0, s266, s65, s178)) :|: s265 >= 0, s265 <= 0, s266 >= 0, s266 <= x'', s177 >= 0, s177 <= x', s178 >= 0, s178 <= x1, s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s67 + 3*s69 + s69*x' + 3*s70 + s70*x'' + 3*s72 + 4*x' + 4*x'' }-> app(if_qs(0, s267, s68, s179), 1 + z'' + if_qs(0, s268, s71, s180)) :|: s267 >= 0, s267 <= 0, s268 >= 0, s268 <= x'', s179 >= 0, s179 <= x', s180 >= 0, s180 <= 0, s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s73 + 3*s75 + s75*x' + 3*s76 + 3*s78 + s78*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s269, s74, s181), 1 + z'' + if_qs(0, s270, s77, s182)) :|: s269 >= 0, s269 <= 0, s270 >= 0, s270 <= 0, s181 >= 0, s181 <= x', s182 >= 0, s182 <= x'', s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s79 + 3*s81 + s81*x' + 3*s82 + 3*s84 + 4*x' }-> app(if_qs(0, s271, s80, s183), 1 + z'' + if_qs(0, s272, s83, s184)) :|: s271 >= 0, s271 <= 0, s272 >= 0, s272 <= 0, s183 >= 0, s183 <= x', s184 >= 0, s184 <= 0, s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s85 + 3*s87 + s87*x' + 3*s88 + 3*s90 + 4*x' }-> app(if_qs(0, s273, s86, s185), 1 + z'' + if_qs(1, s274, s89, s186)) :|: s273 >= 0, s273 <= 0, s274 >= 0, s274 <= 0, s185 >= 0, s185 <= x', s186 >= 0, s186 <= 0, s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 57 + 3*s91 + 3*s93 + 3*s94 + s94*x' + 3*s96 + s96*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s275, s92, s187), 1 + z'' + if_qs(0, s276, s95, s188)) :|: s275 >= 0, s275 <= 0, s276 >= 0, s276 <= x', s187 >= 0, s187 <= 0, s188 >= 0, s188 <= x'', s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s103 + 3*s105 + 3*s106 + s106*x' + 3*s108 + 4*x' }-> app(if_qs(0, s279, s104, s191), 1 + z'' + if_qs(0, s280, s107, s192)) :|: s279 >= 0, s279 <= 0, s280 >= 0, s280 <= x', s191 >= 0, s191 <= 0, s192 >= 0, s192 <= 0, s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s115 + 3*s117 + 3*s118 + 3*s120 + s120*x' + 4*x' }-> app(if_qs(0, s283, s116, s195), 1 + z'' + if_qs(0, s284, s119, s196)) :|: s283 >= 0, s283 <= 0, s284 >= 0, s284 <= 0, s195 >= 0, s195 <= 0, s196 >= 0, s196 <= x', s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 55 + 3*s127 + 3*s129 + 3*s130 + 3*s132 }-> app(if_qs(0, s287, s128, s199), 1 + z'' + if_qs(0, s288, s131, s200)) :|: s287 >= 0, s287 <= 0, s288 >= 0, s288 <= 0, s199 >= 0, s199 <= 0, s200 >= 0, s200 <= 0, s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 55 + 3*s133 + 3*s135 + 3*s136 + 3*s138 }-> app(if_qs(0, s289, s134, s201), 1 + z'' + if_qs(1, s290, s137, s202)) :|: s289 >= 0, s289 <= 0, s290 >= 0, s290 <= 0, s201 >= 0, s201 <= 0, s202 >= 0, s202 <= 0, s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s240 :|: s240 >= 0, s240 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   low(z, z') -{ 42 + 8*x + 8*x' }-> s241 :|: s241 >= 0, s241 <= 1 + (1 + x') + x, x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 43 + 8*x + 2*x*z + 8*x'' + 2*x''*z + 11*z }-> s242 :|: s242 >= 0, s242 <= 1 + (1 + x'') + x, s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 24 }-> if_qs(1, s327, 0, s239) :|: s327 >= 0, s327 <= 0, s239 >= 0, s239 <= 0, z >= 0, z = 0
   quicksort(z) -{ 29 + 3*n + n*x2 + 3*n'' + n''*x3 + 4*x2 + 4*x3 }-> if_qs(0, s295, n', s207) :|: s295 >= 0, s295 <= x2, s207 >= 0, s207 <= x3, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 28 + 3*n + n*x2 + 3*n'' + 4*x2 }-> if_qs(0, s296, n', s208) :|: s296 >= 0, s296 <= x2, s208 >= 0, s208 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 28 + 3*n + 3*n'' + n''*x2 + 4*x2 }-> if_qs(0, s297, n', s209) :|: s297 >= 0, s297 <= 0, s209 >= 0, s209 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + 3*n'' }-> if_qs(0, s298, n', s210) :|: s298 >= 0, s298 <= 0, s210 >= 0, s210 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 28 + 3*n + n*x1 + 4*x1 + 4*x2 }-> if_qs(0, s299, n', s211) :|: s299 >= 0, s299 <= x1, s211 >= 0, s211 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + n*x1 + 4*x1 }-> if_qs(0, s300, n', s212) :|: s300 >= 0, s300 <= x1, s212 >= 0, s212 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n + 4*x1 }-> if_qs(0, s301, n', s213) :|: s301 >= 0, s301 <= 0, s213 >= 0, s213 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n }-> if_qs(0, s302, n', s214) :|: s302 >= 0, s302 <= 0, s214 >= 0, s214 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 28 + 3*n + n*x1 + 3*n' + n'*x2 + 4*x1 + 4*x2 }-> if_qs(0, s303, 0, s215) :|: s303 >= 0, s303 <= x1, s215 >= 0, s215 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + n*x1 + 3*n' + 4*x1 }-> if_qs(0, s304, 0, s216) :|: s304 >= 0, s304 <= x1, s216 >= 0, s216 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, s305, 0, s217) :|: s305 >= 0, s305 <= 0, s217 >= 0, s217 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + 3*n' }-> if_qs(0, s306, 0, s218) :|: s306 >= 0, s306 <= 0, s218 >= 0, s218 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 27 + 3*n + n*x'' + 4*x'' + 4*x1 }-> if_qs(0, s307, 0, s219) :|: s307 >= 0, s307 <= x'', s219 >= 0, s219 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + n*x'' + 4*x'' }-> if_qs(0, s308, 0, s220) :|: s308 >= 0, s308 <= x'', s220 >= 0, s220 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 3*n + 4*x'' }-> if_qs(0, s309, 0, s221) :|: s309 >= 0, s309 <= 0, s221 >= 0, s221 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 3*n }-> if_qs(0, s310, 0, s222) :|: s310 >= 0, s310 <= 0, s222 >= 0, s222 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 28 + 3*n' + n'*x2 + 4*x1 + 4*x2 }-> if_qs(0, s311, n, s223) :|: s311 >= 0, s311 <= x1, s223 >= 0, s223 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n' + 4*x1 }-> if_qs(0, s312, n, s224) :|: s312 >= 0, s312 <= x1, s224 >= 0, s224 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, s313, n, s225) :|: s313 >= 0, s313 <= 0, s225 >= 0, s225 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n' }-> if_qs(0, s314, n, s226) :|: s314 >= 0, s314 <= 0, s226 >= 0, s226 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 27 + 4*x'' + 4*x1 }-> if_qs(0, s315, n, s227) :|: s315 >= 0, s315 <= x'', s227 >= 0, s227 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 4*x'' }-> if_qs(0, s316, n, s228) :|: s316 >= 0, s316 <= x'', s228 >= 0, s228 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 4*x'' }-> if_qs(0, s317, n, s229) :|: s317 >= 0, s317 <= 0, s229 >= 0, s229 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 }-> if_qs(0, s318, n, s230) :|: s318 >= 0, s318 <= 0, s230 >= 0, s230 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 27 + 3*n + n*x1 + 4*x'' + 4*x1 }-> if_qs(0, s319, 0, s231) :|: s319 >= 0, s319 <= x'', s231 >= 0, s231 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + 4*x'' }-> if_qs(0, s320, 0, s232) :|: s320 >= 0, s320 <= x'', s232 >= 0, s232 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 3*n + n*x'' + 4*x'' }-> if_qs(0, s321, 0, s233) :|: s321 >= 0, s321 <= 0, s233 >= 0, s233 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 3*n }-> if_qs(0, s322, 0, s234) :|: s322 >= 0, s322 <= 0, s234 >= 0, s234 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 26 + 4*x' + 4*x'' }-> if_qs(0, s323, 0, s235) :|: s323 >= 0, s323 <= x', s235 >= 0, s235 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 4*x' }-> if_qs(0, s324, 0, s236) :|: s324 >= 0, s324 <= x', s236 >= 0, s236 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 25 + 4*x' }-> if_qs(0, s325, 0, s237) :|: s325 >= 0, s325 <= 0, s237 >= 0, s237 <= x', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 24 }-> if_qs(0, s326, 0, s238) :|: s326 >= 0, s326 <= 0, s238 >= 0, s238 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  if_qs: runtime: ?, size: INF

----------------------------------------

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if_qs
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s151 :|: s151 >= 0, s151 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s152 :|: s152 >= 0, s152 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 42 + 8*x + 8*x1 }-> s153 :|: s153 >= 0, s153 <= 1 + (1 + x1) + x, x1 >= 0, x >= 0, z' = 1 + (1 + x1) + x, z = 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x2 + 2*x2*z + 11*z }-> s154 :|: s154 >= 0, s154 <= 1 + (1 + x2) + x, s'' >= 0, s'' <= 1, z' = 1 + (1 + x2) + x, x >= 0, z - 1 >= 0, x2 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s155 :|: s155 >= 0, s155 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s156 :|: s156 >= 0, s156 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s244 :|: s244 >= 0, s244 <= x, z' >= 0, z'' = 1 + m + x, x >= 0, z = 0, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> 1 + m + s243 :|: s243 >= 0, s243 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   if_qs(z, z', z'', z1) -{ 57 + 3*s100 + s100*x' + 3*s102 + s102*x'' + 3*s97 + 3*s99 + 4*x' + 4*x'' }-> app(if_qs(1, s277, s98, s189), 1 + z'' + if_qs(0, s278, s101, s190)) :|: s277 >= 0, s277 <= 0, s278 >= 0, s278 <= x', s189 >= 0, s189 <= 0, s190 >= 0, s190 <= x'', s97 >= 0, s97 <= z', s98 >= 0, s98 <= z', s99 >= 0, s99 <= z', s100 >= 0, s100 <= z1, s101 >= 0, s101 <= z1, s102 >= 0, s102 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', z' = 0, n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s109 + 3*s111 + 3*s112 + s112*x' + 3*s114 + 4*x' }-> app(if_qs(1, s281, s110, s193), 1 + z'' + if_qs(0, s282, s113, s194)) :|: s281 >= 0, s281 <= 0, s282 >= 0, s282 <= x', s193 >= 0, s193 <= 0, s194 >= 0, s194 <= 0, s109 >= 0, s109 <= z', s110 >= 0, s110 <= z', s111 >= 0, s111 <= z', s112 >= 0, s112 <= z1, s113 >= 0, s113 <= z1, s114 >= 0, s114 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 56 + 3*s121 + 3*s123 + 3*s124 + 3*s126 + s126*x' + 4*x' }-> app(if_qs(1, s285, s122, s197), 1 + z'' + if_qs(0, s286, s125, s198)) :|: s285 >= 0, s285 <= 0, s286 >= 0, s286 <= 0, s197 >= 0, s197 <= 0, s198 >= 0, s198 <= x', s121 >= 0, s121 <= z', s122 >= 0, s122 <= z', s123 >= 0, s123 <= z', s124 >= 0, s124 <= z1, s125 >= 0, s125 <= z1, s126 >= 0, s126 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', z' = 0, n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 55 + 3*s139 + 3*s141 + 3*s142 + 3*s144 }-> app(if_qs(1, s291, s140, s203), 1 + z'' + if_qs(0, s292, s143, s204)) :|: s291 >= 0, s291 <= 0, s292 >= 0, s292 <= 0, s203 >= 0, s203 <= 0, s204 >= 0, s204 <= 0, s139 >= 0, s139 <= z', s140 >= 0, s140 <= z', s141 >= 0, s141 <= z', s142 >= 0, s142 <= z1, s143 >= 0, s143 <= z1, s144 >= 0, s144 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x'
   if_qs(z, z', z'', z1) -{ 55 + 3*s145 + 3*s147 + 3*s148 + 3*s150 }-> app(if_qs(1, s293, s146, s205), 1 + z'' + if_qs(1, s294, s149, s206)) :|: s293 >= 0, s293 <= 0, s294 >= 0, s294 <= 0, s205 >= 0, s205 <= 0, s206 >= 0, s206 <= 0, s145 >= 0, s145 <= z', s146 >= 0, s146 <= z', s147 >= 0, s147 <= z', s148 >= 0, s148 <= z1, s149 >= 0, s149 <= z1, s150 >= 0, s150 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, z' = 0, z1 = 0
   if_qs(z, z', z'', z1) -{ 59 + 3*s1 + s1*x' + 3*s3 + s3*x'' + 3*s4 + s4*x1 + 3*s6 + s6*x2 + 4*x' + 4*x'' + 4*x1 + 4*x2 }-> app(if_qs(0, s245, s2, s157), 1 + z'' + if_qs(0, s246, s5, s158)) :|: s245 >= 0, s245 <= x', s246 >= 0, s246 <= x1, s157 >= 0, s157 <= x'', s158 >= 0, s158 <= x2, s1 >= 0, s1 <= z', s2 >= 0, s2 <= z', s3 >= 0, s3 <= z', s4 >= 0, s4 <= z1, s5 >= 0, s5 <= z1, s6 >= 0, s6 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2, n3 >= 0, x3 >= 0, z' = 1 + n3 + x3, n4 >= 0, x4 >= 0, z1 = 1 + n4 + x4
   if_qs(z, z', z'', z1) -{ 58 + 3*s10 + s10*x1 + 3*s12 + 3*s7 + s7*x' + 3*s9 + s9*x'' + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s247, s8, s159), 1 + z'' + if_qs(0, s248, s11, s160)) :|: s247 >= 0, s247 <= x', s248 >= 0, s248 <= x1, s159 >= 0, s159 <= x'', s160 >= 0, s160 <= 0, s7 >= 0, s7 <= z', s8 >= 0, s8 <= z', s9 >= 0, s9 <= z', s10 >= 0, s10 <= z1, s11 >= 0, s11 <= z1, s12 >= 0, s12 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 58 + 3*s13 + s13*x' + 3*s15 + s15*x'' + 3*s16 + 3*s18 + s18*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s249, s14, s161), 1 + z'' + if_qs(0, s250, s17, s162)) :|: s249 >= 0, s249 <= x', s250 >= 0, s250 <= 0, s161 >= 0, s161 <= x'', s162 >= 0, s162 <= x1, s13 >= 0, s13 <= z', s14 >= 0, s14 <= z', s15 >= 0, s15 <= z', s16 >= 0, s16 <= z1, s17 >= 0, s17 <= z1, s18 >= 0, s18 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s19 + s19*x' + 3*s21 + s21*x'' + 3*s22 + 3*s24 + 4*x' + 4*x'' }-> app(if_qs(0, s251, s20, s163), 1 + z'' + if_qs(0, s252, s23, s164)) :|: s251 >= 0, s251 <= x', s252 >= 0, s252 <= 0, s163 >= 0, s163 <= x'', s164 >= 0, s164 <= 0, s19 >= 0, s19 <= z', s20 >= 0, s20 <= z', s21 >= 0, s21 <= z', s22 >= 0, s22 <= z1, s23 >= 0, s23 <= z1, s24 >= 0, s24 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s25 + s25*x' + 3*s27 + s27*x'' + 3*s28 + 3*s30 + 4*x' + 4*x'' }-> app(if_qs(0, s253, s26, s165), 1 + z'' + if_qs(1, s254, s29, s166)) :|: s253 >= 0, s253 <= x', s254 >= 0, s254 <= 0, s165 >= 0, s165 <= x'', s166 >= 0, s166 <= 0, s25 >= 0, s25 <= z', s26 >= 0, s26 <= z', s27 >= 0, s27 <= z', s28 >= 0, s28 <= z1, s29 >= 0, s29 <= z1, s30 >= 0, s30 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, z1 = 0
   if_qs(z, z', z'', z1) -{ 58 + 3*s31 + s31*x' + 3*s33 + 3*s34 + s34*x'' + 3*s36 + s36*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s255, s32, s167), 1 + z'' + if_qs(0, s256, s35, s168)) :|: s255 >= 0, s255 <= x', s256 >= 0, s256 <= x'', s167 >= 0, s167 <= 0, s168 >= 0, s168 <= x1, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z', s33 >= 0, s33 <= z', s34 >= 0, s34 <= z1, s35 >= 0, s35 <= z1, s36 >= 0, s36 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s37 + s37*x' + 3*s39 + 3*s40 + s40*x'' + 3*s42 + 4*x' + 4*x'' }-> app(if_qs(0, s257, s38, s169), 1 + z'' + if_qs(0, s258, s41, s170)) :|: s257 >= 0, s257 <= x', s258 >= 0, s258 <= x'', s169 >= 0, s169 <= 0, s170 >= 0, s170 <= 0, s37 >= 0, s37 <= z', s38 >= 0, s38 <= z', s39 >= 0, s39 <= z', s40 >= 0, s40 <= z1, s41 >= 0, s41 <= z1, s42 >= 0, s42 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s43 + s43*x' + 3*s45 + 3*s46 + 3*s48 + s48*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s259, s44, s171), 1 + z'' + if_qs(0, s260, s47, s172)) :|: s259 >= 0, s259 <= x', s260 >= 0, s260 <= 0, s171 >= 0, s171 <= 0, s172 >= 0, s172 <= x'', s43 >= 0, s43 <= z', s44 >= 0, s44 <= z', s45 >= 0, s45 <= z', s46 >= 0, s46 <= z1, s47 >= 0, s47 <= z1, s48 >= 0, s48 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s49 + s49*x' + 3*s51 + 3*s52 + 3*s54 + 4*x' }-> app(if_qs(0, s261, s50, s173), 1 + z'' + if_qs(0, s262, s53, s174)) :|: s261 >= 0, s261 <= x', s262 >= 0, s262 <= 0, s173 >= 0, s173 <= 0, s174 >= 0, s174 <= 0, s49 >= 0, s49 <= z', s50 >= 0, s50 <= z', s51 >= 0, s51 <= z', s52 >= 0, s52 <= z1, s53 >= 0, s53 <= z1, s54 >= 0, s54 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s55 + s55*x' + 3*s57 + 3*s58 + 3*s60 + 4*x' }-> app(if_qs(0, s263, s56, s175), 1 + z'' + if_qs(1, s264, s59, s176)) :|: s263 >= 0, s263 <= x', s264 >= 0, s264 <= 0, s175 >= 0, s175 <= 0, s176 >= 0, s176 <= 0, s55 >= 0, s55 <= z', s56 >= 0, s56 <= z', s57 >= 0, s57 <= z', s58 >= 0, s58 <= z1, s59 >= 0, s59 <= z1, s60 >= 0, s60 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 58 + 3*s61 + 3*s63 + s63*x' + 3*s64 + s64*x'' + 3*s66 + s66*x1 + 4*x' + 4*x'' + 4*x1 }-> app(if_qs(0, s265, s62, s177), 1 + z'' + if_qs(0, s266, s65, s178)) :|: s265 >= 0, s265 <= 0, s266 >= 0, s266 <= x'', s177 >= 0, s177 <= x', s178 >= 0, s178 <= x1, s61 >= 0, s61 <= z', s62 >= 0, s62 <= z', s63 >= 0, s63 <= z', s64 >= 0, s64 <= z1, s65 >= 0, s65 <= z1, s66 >= 0, s66 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1, n2 >= 0, x2 >= 0, z' = 1 + n2 + x2, n3 >= 0, x3 >= 0, z1 = 1 + n3 + x3
   if_qs(z, z', z'', z1) -{ 57 + 3*s67 + 3*s69 + s69*x' + 3*s70 + s70*x'' + 3*s72 + 4*x' + 4*x'' }-> app(if_qs(0, s267, s68, s179), 1 + z'' + if_qs(0, s268, s71, s180)) :|: s267 >= 0, s267 <= 0, s268 >= 0, s268 <= x'', s179 >= 0, s179 <= x', s180 >= 0, s180 <= 0, s67 >= 0, s67 <= z', s68 >= 0, s68 <= z', s69 >= 0, s69 <= z', s70 >= 0, s70 <= z1, s71 >= 0, s71 <= z1, s72 >= 0, s72 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 57 + 3*s73 + 3*s75 + s75*x' + 3*s76 + 3*s78 + s78*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s269, s74, s181), 1 + z'' + if_qs(0, s270, s77, s182)) :|: s269 >= 0, s269 <= 0, s270 >= 0, s270 <= 0, s181 >= 0, s181 <= x', s182 >= 0, s182 <= x'', s73 >= 0, s73 <= z', s74 >= 0, s74 <= z', s75 >= 0, s75 <= z', s76 >= 0, s76 <= z1, s77 >= 0, s77 <= z1, s78 >= 0, s78 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s79 + 3*s81 + s81*x' + 3*s82 + 3*s84 + 4*x' }-> app(if_qs(0, s271, s80, s183), 1 + z'' + if_qs(0, s272, s83, s184)) :|: s271 >= 0, s271 <= 0, s272 >= 0, s272 <= 0, s183 >= 0, s183 <= x', s184 >= 0, s184 <= 0, s79 >= 0, s79 <= z', s80 >= 0, s80 <= z', s81 >= 0, s81 <= z', s82 >= 0, s82 <= z1, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s85 + 3*s87 + s87*x' + 3*s88 + 3*s90 + 4*x' }-> app(if_qs(0, s273, s86, s185), 1 + z'' + if_qs(1, s274, s89, s186)) :|: s273 >= 0, s273 <= 0, s274 >= 0, s274 <= 0, s185 >= 0, s185 <= x', s186 >= 0, s186 <= 0, s85 >= 0, s85 <= z', s86 >= 0, s86 <= z', s87 >= 0, s87 <= z', s88 >= 0, s88 <= z1, s89 >= 0, s89 <= z1, s90 >= 0, s90 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', z1 = 0
   if_qs(z, z', z'', z1) -{ 57 + 3*s91 + 3*s93 + 3*s94 + s94*x' + 3*s96 + s96*x'' + 4*x' + 4*x'' }-> app(if_qs(0, s275, s92, s187), 1 + z'' + if_qs(0, s276, s95, s188)) :|: s275 >= 0, s275 <= 0, s276 >= 0, s276 <= x', s187 >= 0, s187 <= 0, s188 >= 0, s188 <= x'', s91 >= 0, s91 <= z', s92 >= 0, s92 <= z', s93 >= 0, s93 <= z', s94 >= 0, s94 <= z1, s95 >= 0, s95 <= z1, s96 >= 0, s96 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x'', n1 >= 0, x1 >= 0, z' = 1 + n1 + x1, n2 >= 0, x2 >= 0, z1 = 1 + n2 + x2
   if_qs(z, z', z'', z1) -{ 56 + 3*s103 + 3*s105 + 3*s106 + s106*x' + 3*s108 + 4*x' }-> app(if_qs(0, s279, s104, s191), 1 + z'' + if_qs(0, s280, s107, s192)) :|: s279 >= 0, s279 <= 0, s280 >= 0, s280 <= x', s191 >= 0, s191 <= 0, s192 >= 0, s192 <= 0, s103 >= 0, s103 <= z', s104 >= 0, s104 <= z', s105 >= 0, s105 <= z', s106 >= 0, s106 <= z1, s107 >= 0, s107 <= z1, s108 >= 0, s108 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 56 + 3*s115 + 3*s117 + 3*s118 + 3*s120 + s120*x' + 4*x' }-> app(if_qs(0, s283, s116, s195), 1 + z'' + if_qs(0, s284, s119, s196)) :|: s283 >= 0, s283 <= 0, s284 >= 0, s284 <= 0, s195 >= 0, s195 <= 0, s196 >= 0, s196 <= x', s115 >= 0, s115 <= z', s116 >= 0, s116 <= z', s117 >= 0, s117 <= z', s118 >= 0, s118 <= z1, s119 >= 0, s119 <= z1, s120 >= 0, s120 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z1 = 1 + n' + x', n'' >= 0, x'' >= 0, z' = 1 + n'' + x'', n1 >= 0, x1 >= 0, z1 = 1 + n1 + x1
   if_qs(z, z', z'', z1) -{ 55 + 3*s127 + 3*s129 + 3*s130 + 3*s132 }-> app(if_qs(0, s287, s128, s199), 1 + z'' + if_qs(0, s288, s131, s200)) :|: s287 >= 0, s287 <= 0, s288 >= 0, s288 <= 0, s199 >= 0, s199 <= 0, s200 >= 0, s200 <= 0, s127 >= 0, s127 <= z', s128 >= 0, s128 <= z', s129 >= 0, s129 <= z', s130 >= 0, s130 <= z1, s131 >= 0, s131 <= z1, s132 >= 0, s132 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', n'' >= 0, x'' >= 0, z1 = 1 + n'' + x''
   if_qs(z, z', z'', z1) -{ 55 + 3*s133 + 3*s135 + 3*s136 + 3*s138 }-> app(if_qs(0, s289, s134, s201), 1 + z'' + if_qs(1, s290, s137, s202)) :|: s289 >= 0, s289 <= 0, s290 >= 0, s290 <= 0, s201 >= 0, s201 <= 0, s202 >= 0, s202 <= 0, s133 >= 0, s133 <= z', s134 >= 0, s134 <= z', s135 >= 0, s135 <= z', s136 >= 0, s136 <= z1, s137 >= 0, s137 <= z1, s138 >= 0, s138 <= z1, z'' >= 0, z' >= 0, z1 >= 0, z = 0, n' >= 0, x' >= 0, z' = 1 + n' + x', z1 = 0
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   isempty(z) -{ 1 }-> 1 :|: z = 0
   isempty(z) -{ 1 }-> 0 :|: n >= 0, x >= 0, z = 1 + n + x
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   low(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s240 :|: s240 >= 0, s240 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   low(z, z') -{ 42 + 8*x + 8*x' }-> s241 :|: s241 >= 0, s241 <= 1 + (1 + x') + x, x' >= 0, x >= 0, z = 0, z' = 1 + (1 + x') + x
   low(z, z') -{ 43 + 8*x + 2*x*z + 8*x'' + 2*x''*z + 11*z }-> s242 :|: s242 >= 0, s242 <= 1 + (1 + x'') + x, s' >= 0, s' <= 1, x >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + x, x'' >= 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   quicksort(z) -{ 24 }-> if_qs(1, s327, 0, s239) :|: s327 >= 0, s327 <= 0, s239 >= 0, s239 <= 0, z >= 0, z = 0
   quicksort(z) -{ 29 + 3*n + n*x2 + 3*n'' + n''*x3 + 4*x2 + 4*x3 }-> if_qs(0, s295, n', s207) :|: s295 >= 0, s295 <= x2, s207 >= 0, s207 <= x3, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3, n3 >= 0, x4 >= 0, z = 1 + n3 + x4
   quicksort(z) -{ 28 + 3*n + n*x2 + 3*n'' + 4*x2 }-> if_qs(0, s296, n', s208) :|: s296 >= 0, s296 <= x2, s208 >= 0, s208 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 28 + 3*n + 3*n'' + n''*x2 + 4*x2 }-> if_qs(0, s297, n', s209) :|: s297 >= 0, s297 <= 0, s209 >= 0, s209 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + 3*n'' }-> if_qs(0, s298, n', s210) :|: s298 >= 0, s298 <= 0, s210 >= 0, s210 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 28 + 3*n + n*x1 + 4*x1 + 4*x2 }-> if_qs(0, s299, n', s211) :|: s299 >= 0, s299 <= x1, s211 >= 0, s211 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + n*x1 + 4*x1 }-> if_qs(0, s300, n', s212) :|: s300 >= 0, s300 <= x1, s212 >= 0, s212 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n + 4*x1 }-> if_qs(0, s301, n', s213) :|: s301 >= 0, s301 <= 0, s213 >= 0, s213 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n }-> if_qs(0, s302, n', s214) :|: s302 >= 0, s302 <= 0, s214 >= 0, s214 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 28 + 3*n + n*x1 + 3*n' + n'*x2 + 4*x1 + 4*x2 }-> if_qs(0, s303, 0, s215) :|: s303 >= 0, s303 <= x1, s215 >= 0, s215 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n + n*x1 + 3*n' + 4*x1 }-> if_qs(0, s304, 0, s216) :|: s304 >= 0, s304 <= x1, s216 >= 0, s216 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, s305, 0, s217) :|: s305 >= 0, s305 <= 0, s217 >= 0, s217 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + 3*n' }-> if_qs(0, s306, 0, s218) :|: s306 >= 0, s306 <= 0, s218 >= 0, s218 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 27 + 3*n + n*x'' + 4*x'' + 4*x1 }-> if_qs(0, s307, 0, s219) :|: s307 >= 0, s307 <= x'', s219 >= 0, s219 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + n*x'' + 4*x'' }-> if_qs(0, s308, 0, s220) :|: s308 >= 0, s308 <= x'', s220 >= 0, s220 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 3*n + 4*x'' }-> if_qs(0, s309, 0, s221) :|: s309 >= 0, s309 <= 0, s221 >= 0, s221 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 3*n }-> if_qs(0, s310, 0, s222) :|: s310 >= 0, s310 <= 0, s222 >= 0, s222 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 28 + 3*n' + n'*x2 + 4*x1 + 4*x2 }-> if_qs(0, s311, n, s223) :|: s311 >= 0, s311 <= x1, s223 >= 0, s223 <= x2, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2, n2 >= 0, x3 >= 0, z = 1 + n2 + x3
   quicksort(z) -{ 27 + 3*n' + 4*x1 }-> if_qs(0, s312, n, s224) :|: s312 >= 0, s312 <= x1, s224 >= 0, s224 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 27 + 3*n' + n'*x1 + 4*x1 }-> if_qs(0, s313, n, s225) :|: s313 >= 0, s313 <= 0, s225 >= 0, s225 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n' }-> if_qs(0, s314, n, s226) :|: s314 >= 0, s314 <= 0, s226 >= 0, s226 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 27 + 4*x'' + 4*x1 }-> if_qs(0, s315, n, s227) :|: s315 >= 0, s315 <= x'', s227 >= 0, s227 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 4*x'' }-> if_qs(0, s316, n, s228) :|: s316 >= 0, s316 <= x'', s228 >= 0, s228 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 4*x'' }-> if_qs(0, s317, n, s229) :|: s317 >= 0, s317 <= 0, s229 >= 0, s229 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 }-> if_qs(0, s318, n, s230) :|: s318 >= 0, s318 <= 0, s230 >= 0, s230 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 27 + 3*n + n*x1 + 4*x'' + 4*x1 }-> if_qs(0, s319, 0, s231) :|: s319 >= 0, s319 <= x'', s231 >= 0, s231 <= x1, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1, n1 >= 0, x2 >= 0, z = 1 + n1 + x2
   quicksort(z) -{ 26 + 3*n + 4*x'' }-> if_qs(0, s320, 0, s232) :|: s320 >= 0, s320 <= x'', s232 >= 0, s232 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 26 + 3*n + n*x'' + 4*x'' }-> if_qs(0, s321, 0, s233) :|: s321 >= 0, s321 <= 0, s233 >= 0, s233 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 3*n }-> if_qs(0, s322, 0, s234) :|: s322 >= 0, s322 <= 0, s234 >= 0, s234 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 26 + 4*x' + 4*x'' }-> if_qs(0, s323, 0, s235) :|: s323 >= 0, s323 <= x', s235 >= 0, s235 <= x'', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x'', n'' >= 0, x1 >= 0, z = 1 + n'' + x1
   quicksort(z) -{ 25 + 4*x' }-> if_qs(0, s324, 0, s236) :|: s324 >= 0, s324 <= x', s236 >= 0, s236 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 25 + 4*x' }-> if_qs(0, s325, 0, s237) :|: s325 >= 0, s325 <= 0, s237 >= 0, s237 <= x', z >= 0, n >= 0, x' >= 0, z = 1 + n + x', n' >= 0, x'' >= 0, z = 1 + n' + x''
   quicksort(z) -{ 24 }-> if_qs(0, s326, 0, s238) :|: s326 >= 0, s326 <= 0, s238 >= 0, s238 <= 0, z >= 0, n >= 0, x' >= 0, z = 1 + n + x'
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if_qs}, {quicksort}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z]
  isempty: runtime: O(1) [1], size: O(1) [1]
  head: runtime: O(1) [1], size: O(n^1) [z]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  high: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_high: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  low: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if_low: runtime: O(n^2) [24 + 6*z' + 2*z'*z'' + 8*z''], size: O(n^1) [z'']
  if_qs: runtime: INF, size: INF

----------------------------------------

(69) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   if_low(v0, v1, v2) -> null_if_low [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   le(v0, v1) -> null_le [0]
   app(v0, v1) -> null_app [0]
   low(v0, v1) -> null_low [0]
   high(v0, v1) -> null_high [0]
   isempty(v0) -> null_isempty [0]
   if_qs(v0, v1, v2, v3) -> null_if_qs [0]

And the following fresh constants:    null_if_low, null_if_high, null_head, null_tail, null_le, null_app, null_low, null_high, null_isempty, null_if_qs

----------------------------------------

(70)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x)) [1]
   if_low(true, n, add(m, x)) -> add(m, low(n, x)) [1]
   if_low(false, n, add(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x)) [1]
   if_high(true, n, add(m, x)) -> high(n, x) [1]
   if_high(false, n, add(m, x)) -> add(m, high(n, x)) [1]
   head(add(n, x)) -> n [1]
   tail(add(n, x)) -> x [1]
   isempty(nil) -> true [1]
   isempty(add(n, x)) -> false [1]
   quicksort(x) -> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) [1]
   if_qs(true, x, n, y) -> nil [1]
   if_qs(false, x, n, y) -> app(quicksort(x), add(n, quicksort(y))) [1]
   if_low(v0, v1, v2) -> null_if_low [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   le(v0, v1) -> null_le [0]
   app(v0, v1) -> null_app [0]
   low(v0, v1) -> null_low [0]
   high(v0, v1) -> null_high [0]
   isempty(v0) -> null_isempty [0]
   if_qs(v0, v1, v2, v3) -> null_if_qs [0]

The TRS has the following type information:
le :: 0:s:null_head -> 0:s:null_head -> true:false:null_le:null_isempty
0 :: 0:s:null_head
true :: true:false:null_le:null_isempty
s :: 0:s:null_head -> 0:s:null_head
false :: true:false:null_le:null_isempty
app :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
nil :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
add :: 0:s:null_head -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
low :: 0:s:null_head -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
if_low :: true:false:null_le:null_isempty -> 0:s:null_head -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
high :: 0:s:null_head -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
if_high :: true:false:null_le:null_isempty -> 0:s:null_head -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
head :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> 0:s:null_head
tail :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
isempty :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> true:false:null_le:null_isempty
quicksort :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
if_qs :: true:false:null_le:null_isempty -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> 0:s:null_head -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs -> nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_if_low :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_if_high :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_head :: 0:s:null_head
null_tail :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_le :: true:false:null_le:null_isempty
null_app :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_low :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_high :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs
null_isempty :: true:false:null_le:null_isempty
null_if_qs :: nil:add:null_if_low:null_if_high:null_tail:null_app:null_low:null_high:null_if_qs

Rewrite Strategy: INNERMOST
----------------------------------------

(71) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 2
   false => 1
   nil => 0
   null_if_low => 0
   null_if_high => 0
   null_head => 0
   null_tail => 0
   null_le => 0
   null_app => 0
   null_low => 0
   null_high => 0
   null_isempty => 0
   null_if_qs => 0

----------------------------------------

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y
   head(z) -{ 1 }-> n :|: n >= 0, x >= 0, z = 1 + n + x
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   high(z, z') -{ 1 }-> if_high(le(m, n), n, 1 + m + x) :|: n >= 0, z' = 1 + m + x, z = n, x >= 0, m >= 0
   high(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   high(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if_high(z, z', z'') -{ 1 }-> high(n, x) :|: z = 2, n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_high(z, z', z'') -{ 1 }-> 1 + m + high(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_low(z, z', z'') -{ 1 }-> low(n, x) :|: n >= 0, z = 1, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_low(z, z', z'') -{ 1 }-> 1 + m + low(n, x) :|: z = 2, n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   if_qs(z, z', z'', z1) -{ 1 }-> app(quicksort(x), 1 + n + quicksort(y)) :|: n >= 0, z1 = y, z' = x, z'' = n, z = 1, x >= 0, y >= 0
   if_qs(z, z', z'', z1) -{ 1 }-> 0 :|: z = 2, n >= 0, z1 = y, z' = x, z'' = n, x >= 0, y >= 0
   if_qs(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   isempty(z) -{ 1 }-> 2 :|: z = 0
   isempty(z) -{ 1 }-> 1 :|: n >= 0, x >= 0, z = 1 + n + x
   isempty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   low(z, z') -{ 1 }-> if_low(le(m, n), n, 1 + m + x) :|: n >= 0, z' = 1 + m + x, z = n, x >= 0, m >= 0
   low(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   low(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quicksort(z) -{ 1 }-> if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) :|: x >= 0, z = x
   tail(z) -{ 1 }-> x :|: n >= 0, x >= 0, z = 1 + n + x
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(73) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(nil, z0) -> c3
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   LOW(z0, nil) -> c5
   LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, nil) -> c9
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   HEAD(add(z0, z1)) -> c13
   TAIL(add(z0, z1)) -> c14
   ISEMPTY(nil) -> c15
   ISEMPTY(add(z0, z1)) -> c16
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), ISEMPTY(z0))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HEAD(z0))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   IF_QS(true, z0, z1, z2) -> c23
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(nil, z0) -> c3
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   LOW(z0, nil) -> c5
   LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, nil) -> c9
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   HEAD(add(z0, z1)) -> c13
   TAIL(add(z0, z1)) -> c14
   ISEMPTY(nil) -> c15
   ISEMPTY(add(z0, z1)) -> c16
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), ISEMPTY(z0))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HEAD(z0))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   IF_QS(true, z0, z1, z2) -> c23
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, HEAD_1, TAIL_1, ISEMPTY_1, QUICKSORT_1, IF_QS_4

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5, c6_2, c7_1, c8_1, c9, c10_2, c11_1, c12_1, c13, c14, c15, c16, c17_2, c18_3, c19_3, c20_2, c21_3, c22_3, c23, c24_2, c25_2


----------------------------------------

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   LOW(z0, nil) -> c5
   APP(nil, z0) -> c3
   ISEMPTY(nil) -> c15
   IF_QS(true, z0, z1, z2) -> c23
   LE(s(z0), 0) -> c1
   HEAD(add(z0, z1)) -> c13
   ISEMPTY(add(z0, z1)) -> c16
   HIGH(z0, nil) -> c9
   TAIL(add(z0, z1)) -> c14
   LE(0, z0) -> c

----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), ISEMPTY(z0))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HEAD(z0))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), ISEMPTY(z0))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HEAD(z0))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c6_2, c7_1, c8_1, c10_2, c11_1, c12_1, c17_2, c18_3, c19_3, c20_2, c21_3, c22_3, c24_2, c25_2


----------------------------------------

(77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c6_2, c7_1, c8_1, c10_2, c11_1, c12_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2


----------------------------------------

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOW(z0, add(z1, z2)) -> c6(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) by 
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)), LE(0, z0))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))

----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)), LE(0, z0))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)), LE(0, z0))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, HIGH_2, IF_HIGH_3, IF_QS_4, QUICKSORT_1, LOW_2

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2


----------------------------------------

(81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, HIGH_2, IF_HIGH_3, IF_QS_4, QUICKSORT_1, LOW_2

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c10_2, c11_1, c12_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1


----------------------------------------

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HIGH(z0, add(z1, z2)) -> c10(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) by 
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)), LE(0, z0))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))

----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)), LE(0, z0))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)), LE(0, z0))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, IF_QS_4, QUICKSORT_1, LOW_2, HIGH_2

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2


----------------------------------------

(85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, IF_QS_4, QUICKSORT_1, LOW_2, HIGH_2

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1


----------------------------------------

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_QS(false, z0, z1, z2) -> c24(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z0)) by 
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))

----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, IF_QS_4, QUICKSORT_1, LOW_2, HIGH_2

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2


----------------------------------------

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_QS(false, z0, z1, z2) -> c25(APP(quicksort(z0), add(z1, quicksort(z2))), QUICKSORT(z2)) by 
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))

----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2


----------------------------------------

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c17(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))) by 
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(nil) -> c17(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(nil) -> c17(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(nil) -> c17(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1


----------------------------------------

(93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c17(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))

----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1


----------------------------------------

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c18(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0))) by 
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c18(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c18(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c18(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2


----------------------------------------

(97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c18(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c19_2, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2


----------------------------------------

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c19(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), LOW(head(z0), tail(z0))) by 
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c19(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c19(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c19(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2


----------------------------------------

(101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c19(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), LOW(head(nil), tail(nil)))

----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c20_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2


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(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c20(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))) by 
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(nil) -> c20(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(nil) -> c20(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(nil) -> c20(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1


----------------------------------------

(105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c20(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))

----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c21_2, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1


----------------------------------------

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c21(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) by 
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c21(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c21(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c21(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2


----------------------------------------

(109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c21(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))

----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, LOW_2, HIGH_2, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c22_2, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2


----------------------------------------

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c22(IF_QS(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) by 
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c22(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c22(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(nil) -> c22(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2


----------------------------------------

(113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c22(IF_QS(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), HIGH(head(nil), tail(nil)))

----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_2, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2


----------------------------------------

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOW(s(z1), add(s(z0), x2)) -> c6(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) by 
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))

----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_2, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2


----------------------------------------

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HIGH(s(z1), add(s(z0), x2)) -> c10(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) by 
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))

----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2


----------------------------------------

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_QS(false, x0, x1, z0) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0)) by 
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))

----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c24_2, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2


----------------------------------------

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(z0)) by 
   IF_QS(false, x0, x1, z0) -> c24(APP(if_qs(isempty(x0), low(head(x0), tail(x0)), head(x0), high(head(x0), tail(x0))), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(x0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(nil))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))

----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(nil))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(nil))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2


----------------------------------------

(123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
K tuples:none
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1


----------------------------------------

(125) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
We considered the (Usable) Rules:
   isempty(add(z0, z1)) -> false
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APP(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF_HIGH(x_1, x_2, x_3)) = x_3
   POL(IF_LOW(x_1, x_2, x_3)) = x_3
   POL(IF_QS(x_1, x_2, x_3, x_4)) = x_1
   POL(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = [1]
   POL(add(x_1, x_2)) = 0
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1, x_2)) = x_1 + x_2
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = [1]
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = 0
   POL(if_high(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(if_low(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(if_qs(x_1, x_2, x_3, x_4)) = 0
   POL(isempty(x_1)) = [1]
   POL(le(x_1, x_2)) = x_2
   POL(low(x_1, x_2)) = 0
   POL(nil) = 0
   POL(quicksort(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = 0
   POL(true) = 0

----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1


----------------------------------------

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_QS(false, x0, x1, z0) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0)) by 
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(nil))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))

----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(nil))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(nil))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1


----------------------------------------

(129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_1


----------------------------------------

(131) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
We considered the (Usable) Rules:
   isempty(add(z0, z1)) -> false
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APP(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF_HIGH(x_1, x_2, x_3)) = x_3
   POL(IF_LOW(x_1, x_2, x_3)) = x_3
   POL(IF_QS(x_1, x_2, x_3, x_4)) = x_1
   POL(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = [1]
   POL(add(x_1, x_2)) = 0
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c17(x_1)) = x_1
   POL(c18(x_1, x_2)) = x_1 + x_2
   POL(c19(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1)) = x_1
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = [1]
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = 0
   POL(if_high(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(if_low(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(if_qs(x_1, x_2, x_3, x_4)) = 0
   POL(isempty(x_1)) = [1]
   POL(le(x_1, x_2)) = x_1 + x_2
   POL(low(x_1, x_2)) = 0
   POL(nil) = 0
   POL(quicksort(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = 0
   POL(true) = 0

----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, IF_QS_4, QUICKSORT_1

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c25_2, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_1


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(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, quicksort(x2))), QUICKSORT(x2)) by 
   IF_QS(false, x0, x1, z0) -> c25(APP(if_qs(isempty(x0), low(head(x0), tail(x0)), head(x0), high(head(x0), tail(x0))), add(x1, if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))

----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1


----------------------------------------

(135) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1))) by 
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))

----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1


----------------------------------------

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))

----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1


----------------------------------------

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1


----------------------------------------

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1


----------------------------------------

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c17_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1


----------------------------------------

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1


----------------------------------------

(147) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1


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(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1


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(151) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1


----------------------------------------

(153) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1


----------------------------------------

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c18_2, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1


----------------------------------------

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2


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(159) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2


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(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2


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(163) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2


----------------------------------------

(165) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2


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(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c19_2, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2


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(169) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2


----------------------------------------

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1))) by 
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))

----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2


----------------------------------------

(173) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))

----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2


----------------------------------------

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2


----------------------------------------

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2


----------------------------------------

(179) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c20_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2


----------------------------------------

(181) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1))))) by 
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))

----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2, c20_1


----------------------------------------

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2, c20_1


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(185) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2, c20_1


----------------------------------------

(187) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2, c20_1


----------------------------------------

(189) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2, c20_1


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(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1)))) by 
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))

----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   head(add(z0, z1)) -> z0
   tail(add(z0, z1)) -> z1
   isempty(nil) -> true
   isempty(add(z0, z1)) -> false
   quicksort(z0) -> if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0)))
   if_qs(true, z0, z1, z2) -> nil
   if_qs(false, z0, z1, z2) -> app(quicksort(z0), add(z1, quicksort(z2)))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   APP(add(z0, z1), z2) -> c4(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c7(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c8(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c11(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c12(HIGH(z0, z2))
   LOW(z0, add(0, x2)) -> c6(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c6(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c10(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c10(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c22(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   LOW(s(z0), add(s(0), x2)) -> c6(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c6(IF_LOW(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c6(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   LOW(s(x0), add(s(x1), x2)) -> c6(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c10(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c10(IF_HIGH(false, s(0), add(s(s(z0)), x2)), LE(s(s(z0)), s(0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_HIGH(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, nil) -> c24(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))), QUICKSORT(x0))
   IF_QS(false, x0, x1, add(z0, z1)) -> c24(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(x0))
   IF_QS(false, z0, x1, x2) -> c24(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(z0))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, add(z0, z1), x1, x2) -> c24(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, x0, x1, add(z0, z1)) -> c25(APP(quicksort(x0), add(x1, if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))), QUICKSORT(add(z0, z1)))
   IF_QS(false, z0, x1, x2) -> c25(APP(if_qs(isempty(z0), low(head(z0), tail(z0)), head(z0), high(head(z0), tail(z0))), add(x1, if_qs(isempty(x2), low(head(x2), tail(x2)), head(x2), high(head(x2), tail(x2))))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, nil, x1, x2) -> c25(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))), QUICKSORT(x2))
   IF_QS(false, add(z0, z1), x1, x2) -> c25(APP(if_qs(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), add(x1, quicksort(x2))), QUICKSORT(x2))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c17(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c17(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c17(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c18(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c18(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c18(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c19(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), LOW(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c19(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), LOW(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c19(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), LOW(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c20(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))))
   QUICKSORT(add(x0, add(z1, z2))) -> c20(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(z0, z1)) -> c20(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), low(head(add(x0, nil)), tail(add(x0, nil))), head(add(x0, nil)), nil), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), low(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), if_high(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), z1)), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), head(add(z0, z1)), high(z0, tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(head(add(z0, z1)), z1), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), tail(add(z0, z1))), z0, high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(x0, nil)) -> c21(IF_QS(isempty(add(x0, nil)), nil, head(add(x0, nil)), high(head(add(x0, nil)), tail(add(x0, nil)))), HIGH(head(add(x0, nil)), tail(add(x0, nil))))
   QUICKSORT(add(x0, add(z1, z2))) -> c21(IF_QS(isempty(add(x0, add(z1, z2))), if_low(le(z1, head(add(x0, add(z1, z2)))), head(add(x0, add(z1, z2))), add(z1, z2)), head(add(x0, add(z1, z2))), high(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2))))), HIGH(head(add(x0, add(z1, z2))), tail(add(x0, add(z1, z2)))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(isempty(add(z0, z1)), low(z0, z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(head(add(z0, z1)), z1), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
   QUICKSORT(add(z0, z1)) -> c21(IF_QS(false, low(z0, tail(add(z0, z1))), head(add(z0, z1)), high(head(add(z0, z1)), tail(add(z0, z1)))), HIGH(head(add(z0, z1)), tail(add(z0, z1))))
K tuples:
   IF_QS(false, nil, x1, x2) -> c24(APP(if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))), add(x1, quicksort(x2))))
   IF_QS(false, x0, x1, nil) -> c25(APP(quicksort(x0), add(x1, if_qs(true, low(head(nil), tail(nil)), head(nil), high(head(nil), tail(nil))))))
Defined Rule Symbols:   le_2, app_2, low_2, if_low_3, high_2, if_high_3, head_1, tail_1, isempty_1, quicksort_1, if_qs_4

Defined Pair Symbols:   LE_2, APP_2, IF_LOW_3, IF_HIGH_3, LOW_2, HIGH_2, QUICKSORT_1, IF_QS_4

Compound Symbols:   c2_1, c4_1, c7_1, c8_1, c11_1, c12_1, c6_1, c10_1, c21_2, c22_2, c6_2, c10_2, c24_2, c24_1, c25_2, c25_1, c17_1, c18_2, c19_2, c20_1