KILLED
proof of input_gqcjpsbM9e.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 10 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 271 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 267 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 234 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 487 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 208 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 1223 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 311 ms]
        (48) CpxRNTS
        (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 1130 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 266 ms]
        (54) CpxRNTS
        (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
        (57) IntTrsBoundProof [UPPER BOUND(ID), 1565 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 224 ms]
        (60) CpxRNTS
    (61) CompletionProof [UPPER BOUND(ID), 0 ms]
    (62) CpxTypedWeightedCompleteTrs
        (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
(65) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(66) CdtProblem
    (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 253 ms]
    (72) CdtProblem
    (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2823 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3087 ms]
    (82) CdtProblem
    (83) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2531 ms]
    (86) CdtProblem
    (87) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2624 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 66 ms]
    (134) CdtProblem
    (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 95 ms]
    (138) CdtProblem
    (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) 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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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))
   quicksort(nil) -> nil
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

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:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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))
   quicksort(nil) -> nil
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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))
   quicksort(nil) -> nil
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   quicksort(nil) -> nil [1]
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   quicksort(nil) -> nil [1]
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
le :: 0:s -> 0:s -> true:false
true :: true:false
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
quicksort :: 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:

   quot_2

(c) The following functions are completely defined:

   minus_2
   le_2
   quicksort_1
   low_2
   high_2
   if_high_3
   if_low_3
   app_2

Due to the following rules being added:

   minus(v0, v1) -> 0 [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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   quicksort(nil) -> nil [1]
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1]
   minus(v0, v1) -> 0 [0]
   if_high(v0, v1, v2) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
le :: 0:s -> 0:s -> true:false
true :: true:false
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
quicksort :: 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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(0)) -> s(quot(x, s(0))) [2]
   quot(s(s(x')), s(s(y'))) -> s(quot(minus(x', y'), s(s(y')))) [2]
   quot(s(x), s(y)) -> s(quot(0, s(y))) [1]
   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(x1), x)) -> if_low(le(x1, y''), s(y''), add(s(x1), 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(x2), x)) -> if_high(false, 0, add(s(x2), x)) [2]
   high(s(y1), add(s(x3), x)) -> if_high(le(x3, y1), s(y1), add(s(x3), 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]
   quicksort(nil) -> nil [1]
   quicksort(add(n, nil)) -> app(quicksort(nil), add(n, quicksort(nil))) [3]
   quicksort(add(n, add(m', x4))) -> app(quicksort(if_low(le(m', n), n, add(m', x4))), add(n, quicksort(if_high(le(m', n), n, add(m', x4))))) [3]
   minus(v0, v1) -> 0 [0]
   if_high(v0, v1, v2) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
le :: 0:s -> 0:s -> true:false
true :: true:false
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
quicksort :: 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
   high(z, z') -{ 2 }-> if_high(le(x3, y1), 1 + y1, 1 + (1 + x3) + x) :|: z = 1 + y1, y1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, y''), 1 + y'', 1 + (1 + x1) + x) :|: z = 1 + y'', x1 >= 0, x >= 0, y'' >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + n + quicksort(0)) :|: z = 1 + n + 0, n >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   quot(z, z') -{ 2 }-> 1 + quot(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x


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

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

(16)
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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0


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

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

   { minus }
   { le }
   { app }
   { quot }
   { high, if_high }
   { low, if_low }
   { quicksort }

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

Function symbols to be analyzed: {minus}, {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort}

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

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

Function symbols to be analyzed: {minus}, {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

Function symbols to be analyzed: {minus}, {le}, {app}, {quot}, {high,if_high}, {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

(27) 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

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

(29) 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'

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

(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
   high(z, z') -{ 2 }-> if_high(le(x3, z - 1), 1 + (z - 1), 1 + (1 + x3) + x) :|: z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + x) :|: x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 3 }-> app(quicksort(if_low(le(m', n), n, 1 + m' + x4)), 1 + n + quicksort(if_high(le(m', n), n, 1 + m' + x4))) :|: n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

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

(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
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

(33) 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'

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

(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
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

(35) 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

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

(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
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0

Function symbols to be analyzed: {high,if_high}, {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z]

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {high,if_high}, {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z]

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

(45) 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''

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {high,if_high}, {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [z]
  high: runtime: ?, size: O(n^1) [z']
  if_high: runtime: ?, size: O(n^1) [z'']

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

(47) 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''

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 3 + z }-> if_high(s2, 1 + (z - 1), 1 + (1 + x3) + x) :|: s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 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 + x2) + x) :|: z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 7 + 2*n }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(if_high(s4, n, 1 + m' + x4))) :|: s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= 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 + s13 :|: s13 >= 0, s13 <= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 39 + 8*m' + 2*m'*n + 10*n + 2*n*x4 + 8*x4 }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(s14)) :|: s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']

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

(51) 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''

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

(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 + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= 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 + s13 :|: s13 >= 0, s13 <= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 39 + 8*m' + 2*m'*n + 10*n + 2*n*x4 + 8*x4 }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(s14)) :|: s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {low,if_low}, {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']

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

(53) 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''

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

(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 + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= 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 + s13 :|: s13 >= 0, s13 <= 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
   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(s1, 1 + (z - 1), 1 + (1 + x1) + x) :|: s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   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, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 39 + 8*m' + 2*m'*n + 10*n + 2*n*x4 + 8*x4 }-> app(quicksort(if_low(s3, n, 1 + m' + x4)), 1 + n + quicksort(s14)) :|: s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']

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

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

(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 + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= 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 + s13 :|: s13 >= 0, s13 <= 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' }-> s19 :|: s19 >= 0, s19 <= 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 + s18 :|: s18 >= 0, s18 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   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' }-> s15 :|: s15 >= 0, s15 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   low(z, z') -{ 42 + 8*x + 8*x'' }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + x, x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 43 + 8*x + 2*x*z + 8*x1 + 2*x1*z + 11*z }-> s17 :|: s17 >= 0, s17 <= 1 + (1 + x1) + x, s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 71 + 16*m' + 4*m'*n + 18*n + 4*n*x4 + 16*x4 }-> app(quicksort(s20), 1 + n + quicksort(s14)) :|: s20 >= 0, s20 <= 1 + m' + x4, s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: quicksort
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

(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 + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= 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 + s13 :|: s13 >= 0, s13 <= 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' }-> s19 :|: s19 >= 0, s19 <= 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 + s18 :|: s18 >= 0, s18 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   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' }-> s15 :|: s15 >= 0, s15 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   low(z, z') -{ 42 + 8*x + 8*x'' }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + x, x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 43 + 8*x + 2*x*z + 8*x1 + 2*x1*z + 11*z }-> s17 :|: s17 >= 0, s17 <= 1 + (1 + x1) + x, s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 71 + 16*m' + 4*m'*n + 18*n + 4*n*x4 + 16*x4 }-> app(quicksort(s20), 1 + n + quicksort(s14)) :|: s20 >= 0, s20 <= 1 + m' + x4, s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']
  quicksort: runtime: ?, size: EXP

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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 + s5 :|: s5 >= 0, s5 <= x + z', n >= 0, x >= 0, z' >= 0, z = 1 + n + x
   high(z, z') -{ 42 + 8*x + 8*x2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + x2) + x, z' = 1 + (1 + x2) + x, x >= 0, z = 0, x2 >= 0
   high(z, z') -{ 43 + 8*x + 2*x*z + 8*x3 + 2*x3*z + 11*z }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + x3) + x, s2 >= 0, s2 <= 1, z - 1 >= 0, x >= 0, z' = 1 + (1 + x3) + x, x3 >= 0
   high(z, z') -{ 26 + 6*z + 2*z*z' + 8*z' }-> s9 :|: s9 >= 0, s9 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   high(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   if_high(z, z', z'') -{ 12 + 4*x + x*z' + 3*z' }-> s12 :|: s12 >= 0, s12 <= 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 + s13 :|: s13 >= 0, s13 <= 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' }-> s19 :|: s19 >= 0, s19 <= 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 + s18 :|: s18 >= 0, s18 <= x, z' >= 0, z = 1, z'' = 1 + m + x, x >= 0, m >= 0
   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' }-> s15 :|: s15 >= 0, s15 <= 1 + 0 + (z' - 1), z >= 0, z' - 1 >= 0
   low(z, z') -{ 42 + 8*x + 8*x'' }-> s16 :|: s16 >= 0, s16 <= 1 + (1 + x'') + x, x >= 0, z' = 1 + (1 + x'') + x, x'' >= 0, z = 0
   low(z, z') -{ 43 + 8*x + 2*x*z + 8*x1 + 2*x1*z + 11*z }-> s17 :|: s17 >= 0, s17 <= 1 + (1 + x1) + x, s1 >= 0, s1 <= 1, x1 >= 0, x >= 0, z - 1 >= 0, z' = 1 + (1 + x1) + x
   low(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   quicksort(z) -{ 71 + 16*m' + 4*m'*n + 18*n + 4*n*x4 + 16*x4 }-> app(quicksort(s20), 1 + n + quicksort(s14)) :|: s20 >= 0, s20 <= 1 + m' + x4, s14 >= 0, s14 <= 1 + m' + x4, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 1, n >= 0, x4 >= 0, m' >= 0, z = 1 + n + (1 + m' + x4)
   quicksort(z) -{ 3 }-> app(quicksort(0), 1 + (z - 1) + quicksort(0)) :|: z - 1 >= 0
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 4 + 3*z }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' = 1 + 0
   quot(z, z') -{ 5 + 2*s' + s'*z' + 2*z' }-> 1 + s7 :|: s7 >= 0, s7 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   quot(z, z') -{ 5 + z' }-> 1 + s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' - 1 >= 0

Function symbols to be analyzed: {quicksort}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  quot: runtime: O(n^2) [4 + 2*z + z*z' + z'], size: O(n^1) [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'']
  quicksort: runtime: INF, size: EXP

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

(61) 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:

   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   if_high(v0, v1, v2) -> null_if_high [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]
   quicksort(v0) -> null_quicksort [0]

And the following fresh constants:    null_minus, null_quot, null_if_low, null_if_high, null_le, null_app, null_low, null_high, null_quicksort

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

(62)
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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   quicksort(nil) -> nil [1]
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x)))) [1]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   if_high(v0, v1, v2) -> null_if_high [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]
   quicksort(v0) -> null_quicksort [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
0 :: 0:s:null_minus:null_quot
s :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
quot :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
le :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> true:false:null_le
true :: true:false:null_le
false :: true:false:null_le
app :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
nil :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
add :: 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
low :: 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
if_low :: true:false:null_le -> 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
high :: 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
if_high :: true:false:null_le -> 0:s:null_minus:null_quot -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
quicksort :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort -> nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
null_minus :: 0:s:null_minus:null_quot
null_quot :: 0:s:null_minus:null_quot
null_if_low :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
null_if_high :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
null_le :: true:false:null_le
null_app :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
null_low :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
null_high :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort
null_quicksort :: nil:add:null_if_low:null_if_high:null_app:null_low:null_high:null_quicksort

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

(63) 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_minus => 0
   null_quot => 0
   null_if_low => 0
   null_if_high => 0
   null_le => 0
   null_app => 0
   null_low => 0
   null_high => 0
   null_quicksort => 0

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

(64)
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
   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
   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
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quicksort(z) -{ 1 }-> app(quicksort(low(n, x)), 1 + n + quicksort(high(n, x))) :|: n >= 0, x >= 0, z = 1 + n + x
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   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))
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
Tuples:
   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c2
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(0, z0) -> c4
   LE(s(z0), 0) -> c5
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(nil, z0) -> c7
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, nil) -> c9
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, nil) -> c13
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(nil) -> c17
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
S tuples:
   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c2
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(0, z0) -> c4
   LE(s(z0), 0) -> c5
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(nil, z0) -> c7
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, nil) -> c9
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, nil) -> c13
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(nil) -> c17
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, quot_2, le_2, app_2, low_2, if_low_3, high_2, if_high_3, quicksort_1

Defined Pair Symbols:   MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5, c6_1, c7, c8_1, c9, c10_2, c11_1, c12_1, c13, c14_2, c15_1, c16_1, c17, c18_3, c19_3


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

(67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   APP(nil, z0) -> c7
   QUOT(0, s(z0)) -> c2
   MINUS(z0, 0) -> c
   LE(0, z0) -> c4
   HIGH(z0, nil) -> c13
   QUICKSORT(nil) -> c17
   LOW(z0, nil) -> c9
   LE(s(z0), 0) -> c5

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   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))
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, quot_2, le_2, app_2, low_2, if_low_3, high_2, if_high_3, quicksort_1

Defined Pair Symbols:   MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3


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

(69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3


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

(71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
We considered the (Usable) Rules:
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(z0, 0) -> z0
And the Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   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(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(MINUS(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = 0
   POL(QUOT(x_1, x_2)) = x_1
   POL(add(x_1, x_2)) = 0
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c1(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(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = 0
   POL(high(x_1, x_2)) = x_1 + x_2
   POL(if_high(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if_low(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(le(x_1, x_2)) = x_1 + x_2
   POL(low(x_1, x_2)) = x_1 + x_2
   POL(minus(x_1, x_2)) = x_1
   POL(nil) = [1]
   POL(quicksort(x_1)) = 0
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3


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

(73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
We considered the (Usable) Rules:
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   low(z0, nil) -> nil
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   minus(s(z0), s(z1)) -> minus(z0, z1)
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   minus(z0, 0) -> z0
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
And the Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(APP(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(MINUS(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = x_1
   POL(QUOT(x_1, x_2)) = x_1
   POL(add(x_1, x_2)) = [1] + x_2
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c1(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(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(le(x_1, x_2)) = x_1 + x_2
   POL(low(x_1, x_2)) = x_2
   POL(minus(x_1, x_2)) = [1] + x_1
   POL(nil) = [1]
   POL(quicksort(x_1)) = 0
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, QUOT_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) by 
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)), MINUS(z0, 0))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)), MINUS(z0, 0))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
We considered the (Usable) Rules:
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(z0, 0) -> z0
And the Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
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)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(MINUS(x_1, x_2)) = [2] + x_2
   POL(QUICKSORT(x_1)) = 0
   POL(QUOT(x_1, x_2)) = x_1*x_2
   POL(add(x_1, x_2)) = 0
   POL(app(x_1, x_2)) = [1]
   POL(c1(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(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = 0
   POL(high(x_1, x_2)) = 0
   POL(if_high(x_1, x_2, x_3)) = [1] + x_2 + x_2^2
   POL(if_low(x_1, x_2, x_3)) = [1] + x_2 + x_2^2
   POL(le(x_1, x_2)) = 0
   POL(low(x_1, x_2)) = 0
   POL(minus(x_1, x_2)) = [1] + x_1
   POL(nil) = 0
   POL(quicksort(x_1)) = 0
   POL(s(x_1)) = [2] + x_1
   POL(true) = 0

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
We considered the (Usable) Rules:
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   low(z0, nil) -> nil
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
And the Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
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)) = x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = x_3
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(MINUS(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = [2]x_1^2
   POL(QUOT(x_1, x_2)) = 0
   POL(add(x_1, x_2)) = [1] + x_2
   POL(app(x_1, x_2)) = [2]
   POL(c1(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(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(le(x_1, x_2)) = 0
   POL(low(x_1, x_2)) = x_2
   POL(minus(x_1, x_2)) = 0
   POL(nil) = 0
   POL(quicksort(x_1)) = 0
   POL(s(x_1)) = 0
   POL(true) = 0

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

(83) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

(85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
We considered the (Usable) Rules:
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   low(z0, nil) -> nil
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
And the Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
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)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = x_3
   POL(LE(x_1, x_2)) = 0
   POL(LOW(x_1, x_2)) = x_2
   POL(MINUS(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = x_1^2
   POL(QUOT(x_1, x_2)) = 0
   POL(add(x_1, x_2)) = [2] + x_2
   POL(app(x_1, x_2)) = [2]
   POL(c1(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(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(le(x_1, x_2)) = 0
   POL(low(x_1, x_2)) = x_2
   POL(minus(x_1, x_2)) = 0
   POL(nil) = 0
   POL(quicksort(x_1)) = 0
   POL(s(x_1)) = 0
   POL(true) = 0

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

(87) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

(89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
We considered the (Usable) Rules:
   low(z0, add(z1, z2)) -> if_low(le(z1, z0), z0, add(z1, z2))
   low(z0, nil) -> nil
   if_high(false, z0, add(z1, z2)) -> add(z1, high(z0, z2))
   le(s(z0), s(z1)) -> le(z0, z1)
   if_low(true, z0, add(z1, z2)) -> add(z1, low(z0, z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   high(z0, nil) -> nil
   high(z0, add(z1, z2)) -> if_high(le(z1, z0), z0, add(z1, z2))
   if_low(false, z0, add(z1, z2)) -> low(z0, z2)
   if_high(true, z0, add(z1, z2)) -> high(z0, z2)
And the Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
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)) = [2]x_1 + [2]x_1*x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = [2]x_2*x_3
   POL(IF_LOW(x_1, x_2, x_3)) = x_2*x_3
   POL(LE(x_1, x_2)) = [2]x_2
   POL(LOW(x_1, x_2)) = [2]x_1 + x_1*x_2
   POL(MINUS(x_1, x_2)) = 0
   POL(QUICKSORT(x_1)) = [2]x_1^2
   POL(QUOT(x_1, x_2)) = 0
   POL(add(x_1, x_2)) = [2] + x_1 + x_2
   POL(app(x_1, x_2)) = [2]
   POL(c1(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(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c18(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c19(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = [1]
   POL(high(x_1, x_2)) = [2] + x_2
   POL(if_high(x_1, x_2, x_3)) = [2] + x_1*x_3
   POL(if_low(x_1, x_2, x_3)) = [2] + x_3
   POL(le(x_1, x_2)) = [1]
   POL(low(x_1, x_2)) = [2] + x_2
   POL(minus(x_1, x_2)) = 0
   POL(nil) = 0
   POL(quicksort(x_1)) = 0
   POL(s(x_1)) = [2] + x_1
   POL(true) = [1]

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2

Compound Symbols:   c1_1, c6_1, c8_1, c10_2, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1


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

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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)), LE(0, z0))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2


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

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c14_2, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2, c10_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0)) by 
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)), LE(0, z0))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)), LE(0, z0))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)), LE(s(z0), 0))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2


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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c18_3, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1)) by 
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)), LOW(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)), LOW(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)), LOW(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)), LOW(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3


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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUICKSORT_1, QUOT_2, LOW_2, HIGH_2

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c19_3, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1)) by 
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)), HIGH(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)), HIGH(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)), HIGH(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)), HIGH(z0, nil))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3


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

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1)))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_2, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3, c19_2


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUOT(s(s(z0)), s(s(z1))) -> c3(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) by 
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_2, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOW(s(z1), add(s(z0), x2)) -> c10(IF_LOW(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) by 
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_2, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2, c10_2


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HIGH(s(z1), add(s(z0), x2)) -> c14(IF_HIGH(le(z0, z1), s(z1), add(s(z0), x2)), LE(s(z0), s(z1))) by 
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by 
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_3, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by 
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by 
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c18_2, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(low(z0, nil))) by 
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(low(x0, nil)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(low(x0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2


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

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(low(z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) by 
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_3, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(high(z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) by 
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3


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

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(low(z0, nil)), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil))) by 
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c19_2, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3


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

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(high(z0, nil)))), QUICKSORT(high(z0, nil))) by 
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c19(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(high(x0, nil)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c19(APP(nil, add(x0, quicksort(high(x0, nil)))), QUICKSORT(high(x0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3, c19_2


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

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c18_1, c19_3, c19_2, c19_1


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(x0, nil)) -> c18(QUICKSORT(low(x0, nil))) by 
   QUICKSORT(add(z0, nil)) -> c18(QUICKSORT(nil))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(QUICKSORT(nil))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   QUICKSORT(add(z0, z1)) -> c18(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(low(z0, z1)), LOW(z0, z1))
   QUICKSORT(add(z0, z1)) -> c19(APP(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1)))), QUICKSORT(high(z0, z1)), HIGH(z0, z1))
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   HIGH(z0, add(z1, z2)) -> c14(IF_HIGH(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LOW(z0, add(z1, z2)) -> c10(IF_LOW(le(z1, z0), z0, add(z1, z2)), LE(z1, z0))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c19_1, c18_1


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

(135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(add(z0, nil)) -> c18(QUICKSORT(nil))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c19_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(add(x0, nil)) -> c19(QUICKSORT(high(x0, nil))) by 
   QUICKSORT(add(z0, nil)) -> c19(QUICKSORT(nil))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(QUICKSORT(nil))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c19_1


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

(139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(add(z0, nil)) -> c19(QUICKSORT(nil))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c11_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2


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

(141) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF_LOW(true, z0, add(z1, z2)) -> c11(LOW(z0, z2)) by 
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1


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

(143) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) by QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) by QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) by QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_LOW_3, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c1_1, c6_1, c8_1, c12_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1


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

(149) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF_LOW(false, z0, add(z1, z2)) -> c12(LOW(z0, z2)) by 
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3

Compound Symbols:   c1_1, c6_1, c8_1, c15_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1


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

(151) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF_HIGH(true, z0, add(z1, z2)) -> c15(HIGH(z0, z2)) by 
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, IF_HIGH_3, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3

Compound Symbols:   c1_1, c6_1, c8_1, c16_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1


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

(153) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF_HIGH(false, z0, add(z1, z2)) -> c16(HIGH(z0, z2)) by 
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3

Compound Symbols:   c1_1, c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1


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

(155) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace QUOT(s(s(x0)), s(s(x1))) -> c3(MINUS(s(x0), s(x1))) by 
   QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0)))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0)))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3

Compound Symbols:   c1_1, c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1


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(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0)))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   MINUS_2, LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3

Compound Symbols:   c1_1, c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1


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(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) by 
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0)))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2

Compound Symbols:   c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1


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

(161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUOT(s(s(z0)), s(s(0))) -> c3(MINUS(s(z0), s(0)))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))), MINUS(s(z0), s(0)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2

Compound Symbols:   c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1


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

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2

Compound Symbols:   c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(low(z0, add(z1, z2))), LOW(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2

Compound Symbols:   c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1


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

(167) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, add(0, x2))) -> c18(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(low(z0, add(0, x2))), LOW(z0, add(0, x2))) by QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LE_2, APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2

Compound Symbols:   c6_1, c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(z0), s(z1)) -> c6(LE(z0, z1)) by 
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(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)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)), LE(s(0), s(z0)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(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)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2

Compound Symbols:   c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1


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

(171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
S tuples:
   APP(add(z0, z1), z2) -> c8(APP(z1, z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   APP_2, QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2

Compound Symbols:   c8_1, c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1


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

(173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APP(add(z0, z1), z2) -> c8(APP(z1, z2)) by 
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1, c8_1


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

(175) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(0, add(s(z0), x2))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(low(0, add(s(z0), x2))), LOW(0, add(s(z0), x2))) by QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0)))
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   QUOT_2, LOW_2, HIGH_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c3_1, c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c1_1, c6_1, c8_1


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

(177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QUOT(s(z0), s(0)) -> c3(QUOT(z0, s(0))) by 
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c18_3, c18_2, c19_3, c19_2, c11_1, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(s(z1), add(s(z0), x2))) -> c18(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(low(s(z1), add(s(z0), x2))), LOW(s(z1), add(s(z0), x2))) by QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1


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

(181) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(x0, nil)) -> c18(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(low(x0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)), QUICKSORT(nil))
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)), QUICKSORT(nil))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1


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

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil))
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c18_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(low(z0, nil))) by QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(nil))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1, c18_2


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

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(low(z0, add(0, x2))), add(z0, quicksort(if_high(true, z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2))) by QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1)))
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
   QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(low(0, add(s(z0), x2))), add(0, quicksort(if_high(false, 0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2))) by QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1)))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
   QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(197) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(low(s(z1), add(s(z0), x2))), add(s(z1), quicksort(if_high(le(z0, z1), s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2))) by QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2)))
----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
   QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(199) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2))) by QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_high(le(z1, z0), z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(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)) -> c10(LE(s(x1), s(x0)))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
   QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_high(le(z1, z0), z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1


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

(201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOW(s(x0), add(s(x1), x2)) -> c10(LE(s(x1), s(x0))) by 
   LOW(s(s(y1)), add(s(s(y0)), z2)) -> c10(LE(s(s(y0)), s(s(y1))))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   quicksort(nil) -> nil
   quicksort(add(z0, z1)) -> app(quicksort(low(z0, z1)), add(z0, quicksort(high(z0, z1))))
   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)
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, 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))
Tuples:
   LOW(z0, add(0, x2)) -> c10(IF_LOW(true, z0, add(0, x2)))
   LOW(0, add(s(z0), x2)) -> c10(IF_LOW(false, 0, add(s(z0), x2)))
   HIGH(z0, add(0, x2)) -> c14(IF_HIGH(true, z0, add(0, x2)))
   HIGH(0, add(s(z0), x2)) -> c14(IF_HIGH(false, 0, add(s(z0), x2)))
   QUOT(s(s(s(z0))), s(s(s(z1)))) -> c3(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1))))
   LOW(s(s(z1)), add(s(s(z0)), x2)) -> c10(IF_LOW(le(z0, z1), s(s(z1)), add(s(s(z0)), x2)), LE(s(s(z0)), s(s(z1))))
   HIGH(s(s(z1)), add(s(s(z0)), x2)) -> c14(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)) -> c14(LE(s(x1), s(x0)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(high(z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   QUICKSORT(add(z0, add(0, x2))) -> c19(APP(quicksort(if_low(true, z0, add(0, x2))), add(z0, quicksort(high(z0, add(0, x2))))), QUICKSORT(high(z0, add(0, x2))), HIGH(z0, add(0, x2)))
   QUICKSORT(add(0, add(s(z0), x2))) -> c19(APP(quicksort(if_low(false, 0, add(s(z0), x2))), add(0, quicksort(high(0, add(s(z0), x2))))), QUICKSORT(high(0, add(s(z0), x2))), HIGH(0, add(s(z0), x2)))
   QUICKSORT(add(s(z1), add(s(z0), x2))) -> c19(APP(quicksort(if_low(le(z0, z1), s(z1), add(s(z0), x2))), add(s(z1), quicksort(high(s(z1), add(s(z0), x2))))), QUICKSORT(high(s(z1), add(s(z0), x2))), HIGH(s(z1), add(s(z0), x2)))
   QUICKSORT(add(x0, nil)) -> c19(APP(quicksort(low(x0, nil)), add(x0, nil)), QUICKSORT(high(x0, nil)))
   QUICKSORT(add(z0, nil)) -> c19(APP(quicksort(nil), add(z0, quicksort(nil))), QUICKSORT(high(z0, nil)))
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   QUOT(s(s(z0)), s(s(s(x1)))) -> c3(MINUS(s(z0), s(s(x1))))
   QUICKSORT(add(z0, add(z1, z2))) -> c18(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_low(le(z1, z0), z0, add(z1, z2))), LOW(z0, add(z1, z2)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   QUOT(s(s(z0)), s(s(0))) -> c3(QUOT(z0, s(s(0))))
   QUICKSORT(add(z0, add(0, z1))) -> c18(APP(quicksort(if_low(true, z0, add(0, z1))), add(z0, quicksort(high(z0, add(0, z1))))), QUICKSORT(if_low(le(0, z0), z0, add(0, z1))), LOW(z0, add(0, z1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
   LOW(s(z0), add(s(0), x2)) -> c10(IF_LOW(true, s(z0), add(s(0), x2)))
   LOW(s(0), add(s(s(z0)), x2)) -> c10(IF_LOW(false, s(0), add(s(s(z0)), x2)))
   HIGH(s(z0), add(s(0), x2)) -> c14(IF_HIGH(true, s(z0), add(s(0), x2)))
   HIGH(s(0), add(s(s(z0)), x2)) -> c14(IF_HIGH(false, s(0), add(s(s(z0)), x2)))
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
   QUICKSORT(add(0, add(s(z0), z1))) -> c18(APP(quicksort(if_low(false, 0, add(s(z0), z1))), add(0, quicksort(high(0, add(s(z0), z1))))), QUICKSORT(if_low(le(s(z0), 0), 0, add(s(z0), z1))), LOW(0, add(s(z0), z1)))
   QUOT(s(s(y0)), s(0)) -> c3(QUOT(s(y0), s(0)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c18(APP(quicksort(if_low(le(z1, z0), s(z0), add(s(z1), z2))), add(s(z0), quicksort(high(s(z0), add(s(z1), z2))))), QUICKSORT(if_low(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), LOW(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(low(z0, nil)), add(z0, nil)))
   QUICKSORT(add(z0, nil)) -> c18(APP(quicksort(nil), add(z0, quicksort(nil))))
   QUICKSORT(add(z0, add(0, z1))) -> c19(APP(quicksort(low(z0, add(0, z1))), add(z0, quicksort(if_high(true, z0, add(0, z1))))), QUICKSORT(if_high(le(0, z0), z0, add(0, z1))), HIGH(z0, add(0, z1)))
   QUICKSORT(add(0, add(s(z0), z1))) -> c19(APP(quicksort(low(0, add(s(z0), z1))), add(0, quicksort(if_high(false, 0, add(s(z0), z1))))), QUICKSORT(if_high(le(s(z0), 0), 0, add(s(z0), z1))), HIGH(0, add(s(z0), z1)))
   QUICKSORT(add(s(z0), add(s(z1), z2))) -> c19(APP(quicksort(low(s(z0), add(s(z1), z2))), add(s(z0), quicksort(if_high(le(z1, z0), s(z0), add(s(z1), z2))))), QUICKSORT(if_high(le(s(z1), s(z0)), s(z0), add(s(z1), z2))), HIGH(s(z0), add(s(z1), z2)))
   QUICKSORT(add(z0, add(z1, z2))) -> c19(APP(quicksort(if_low(le(z1, z0), z0, add(z1, z2))), add(z0, quicksort(if_high(le(z1, z0), z0, add(z1, z2))))), QUICKSORT(if_high(le(z1, z0), z0, add(z1, z2))), HIGH(z0, add(z1, z2)))
   LOW(s(s(y1)), add(s(s(y0)), z2)) -> c10(LE(s(s(y0)), s(s(y1))))
S tuples:
   APP(add(z0, add(y0, y1)), z2) -> c8(APP(add(y0, y1), z2))
K tuples:
   IF_LOW(true, x0, add(0, x1)) -> c11(LOW(x0, x1))
   IF_LOW(true, s(x0), add(s(0), x1)) -> c11(LOW(s(x0), x1))
   IF_LOW(true, s(s(x0)), add(s(s(x1)), x2)) -> c11(LOW(s(s(x0)), x2))
   IF_LOW(false, 0, add(s(x0), x1)) -> c12(LOW(0, x1))
   IF_LOW(false, s(0), add(s(s(x0)), x1)) -> c12(LOW(s(0), x1))
   IF_LOW(false, s(s(x0)), add(s(s(x1)), x2)) -> c12(LOW(s(s(x0)), x2))
   IF_HIGH(true, x0, add(0, x1)) -> c15(HIGH(x0, x1))
   IF_HIGH(true, s(x0), add(s(0), x1)) -> c15(HIGH(s(x0), x1))
   IF_HIGH(true, s(s(x0)), add(s(s(x1)), x2)) -> c15(HIGH(s(s(x0)), x2))
   IF_HIGH(false, 0, add(s(x0), x1)) -> c16(HIGH(0, x1))
   IF_HIGH(false, s(0), add(s(s(x0)), x1)) -> c16(HIGH(s(0), x1))
   IF_HIGH(false, s(s(x0)), add(s(s(x1)), x2)) -> c16(HIGH(s(s(x0)), x2))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1)))
Defined Rule Symbols:   minus_2, le_2, quicksort_1, low_2, if_low_3, app_2, high_2, if_high_3

Defined Pair Symbols:   LOW_2, HIGH_2, QUOT_2, QUICKSORT_1, IF_LOW_3, IF_HIGH_3, MINUS_2, LE_2, APP_2

Compound Symbols:   c10_1, c14_1, c3_2, c10_2, c14_2, c19_3, c19_2, c11_1, c18_3, c12_1, c15_1, c16_1, c3_1, c1_1, c6_1, c8_1, c18_1