KILLED
proof of input_XiAmXpwWp6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 18 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 17 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 13 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), 402 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 325 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 164 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 1031 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 417 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 1043 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 410 ms]
        (48) CpxRNTS
        (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 6650 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 3253 ms]
        (54) CpxRNTS
    (55) CompletionProof [UPPER BOUND(ID), 0 ms]
    (56) CpxTypedWeightedCompleteTrs
        (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 18 ms]
        (58) CpxRNTS
(59) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(60) CdtProblem
    (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 205 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 94 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 106 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 106 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 3 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 103 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 132 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtInstantiationProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem


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

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


The TRS R consists of the following rules:

   qsort(nil) -> nil
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
   filterlow(n, nil) -> nil
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
   if1(true, n, x, xs) -> filterlow(n, xs)
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
   filterhigh(n, nil) -> nil
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
   if2(true, n, x, xs) -> filterhigh(n, xs)
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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:

   qsort(nil) -> nil
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
   filterlow(n, nil) -> nil
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
   if1(true, n, x, xs) -> filterlow(n, xs)
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
   filterhigh(n, nil) -> nil
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
   if2(true, n, x, xs) -> filterhigh(n, xs)
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
   last(nil) -> 0'
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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:

   qsort(nil) -> nil
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
   filterlow(n, nil) -> nil
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
   if1(true, n, x, xs) -> filterlow(n, xs)
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
   filterhigh(n, nil) -> nil
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
   if2(true, n, x, xs) -> filterhigh(n, xs)
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   last(nil) -> 0 [1]
   last(cons(x, nil)) -> x [1]
   last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   last(nil) -> 0 [1]
   last(cons(x, nil)) -> x [1]
   last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1]

The TRS has the following type information:
qsort :: nil:cons:ys -> nil:cons:ys
nil :: nil:cons:ys
cons :: 0:s -> nil:cons:ys -> nil:cons:ys
append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys
filterlow :: 0:s -> nil:cons:ys -> nil:cons:ys
last :: nil:cons:ys -> 0:s
filterhigh :: 0:s -> nil:cons:ys -> nil:cons:ys
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys

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

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

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

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

(c) The following functions are completely defined:

   ge_2
   qsort_1
   filterlow_2
   last_1
   filterhigh_2
   if1_4
   if2_4
   append_2

Due to the following rules being added:

   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   last(v0) -> 0 [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

And the following fresh constants:    null_qsort, null_filterlow, null_filterhigh, null_append

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

(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:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   last(nil) -> 0 [1]
   last(cons(x, nil)) -> x [1]
   last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1]
   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   last(v0) -> 0 [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

The TRS has the following type information:
qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
last :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> 0:s
filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append

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:

   qsort(nil) -> nil [1]
   qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(x, qsort(filterhigh(x, cons(x, nil))))) [4]
   qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(x, qsort(filterhigh(0, cons(x, nil))))) [3]
   qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(0, qsort(filterhigh(x, cons(x, nil))))) [3]
   qsort(cons(x, nil)) -> append(qsort(filterlow(x, cons(x, nil))), cons(0, qsort(filterhigh(0, cons(x, nil))))) [2]
   qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(last(cons(y', xs')), qsort(filterhigh(last(cons(y', xs')), cons(x, cons(y', xs')))))) [4]
   qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(last(cons(y', xs')), qsort(filterhigh(0, cons(x, cons(y', xs')))))) [3]
   qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(0, qsort(filterhigh(last(cons(y', xs')), cons(x, cons(y', xs')))))) [3]
   qsort(cons(x, cons(y', xs'))) -> append(qsort(filterlow(last(cons(y', xs')), cons(x, cons(y', xs')))), cons(0, qsort(filterhigh(0, cons(x, cons(y', xs')))))) [2]
   qsort(cons(x, nil)) -> append(qsort(filterlow(0, cons(x, nil))), cons(x, qsort(filterhigh(x, cons(x, nil))))) [3]
   qsort(cons(x, nil)) -> append(qsort(filterlow(0, cons(x, nil))), cons(x, qsort(filterhigh(0, cons(x, nil))))) [2]
   qsort(cons(x, cons(y'', xs''))) -> append(qsort(filterlow(0, cons(x, cons(y'', xs'')))), cons(last(cons(y'', xs'')), qsort(filterhigh(last(cons(y'', xs'')), cons(x, cons(y'', xs'')))))) [3]
   qsort(cons(x, cons(y'', xs''))) -> append(qsort(filterlow(0, cons(x, cons(y'', xs'')))), cons(last(cons(y'', xs'')), qsort(filterhigh(0, cons(x, cons(y'', xs'')))))) [2]
   qsort(cons(x, nil)) -> append(qsort(filterlow(0, cons(x, nil))), cons(0, qsort(filterhigh(x, cons(x, nil))))) [2]
   qsort(cons(x, cons(y1, xs1))) -> append(qsort(filterlow(0, cons(x, cons(y1, xs1)))), cons(0, qsort(filterhigh(last(cons(y1, xs1)), cons(x, cons(y1, xs1)))))) [2]
   qsort(cons(x, xs)) -> append(qsort(filterlow(0, cons(x, xs))), cons(0, qsort(filterhigh(0, cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(0, xs)) -> if1(true, n, 0, xs) [2]
   filterlow(0, cons(s(x'), xs)) -> if1(false, 0, s(x'), xs) [2]
   filterlow(s(x''), cons(s(y2), xs)) -> if1(ge(x'', y2), s(x''), s(y2), xs) [2]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(0, cons(x, xs)) -> if2(true, 0, x, xs) [2]
   filterhigh(s(x1), cons(0, xs)) -> if2(false, s(x1), 0, xs) [2]
   filterhigh(s(y3), cons(s(x2), xs)) -> if2(ge(x2, y3), s(y3), s(x2), xs) [2]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   last(nil) -> 0 [1]
   last(cons(x, nil)) -> x [1]
   last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1]
   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   last(v0) -> 0 [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]

The TRS has the following type information:
qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
cons :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
filterlow :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
last :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> 0:s
filterhigh :: 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
0 :: 0:s
s :: 0:s -> 0:s
ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append
null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append

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:

   nil => 0
   true => 1
   false => 0
   0 => 0
   ys => 1
   null_qsort => 0
   null_filterlow => 0
   null_filterhigh => 0
   null_append => 0

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, y3), 1 + y3, 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 1 + y3, y3 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + x1, 0, xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + x1
   filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   filterlow(z, z') -{ 2 }-> if1(ge(x'', y2), 1 + x'', 1 + y2, xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y2) + xs, x'' >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, n, 0, xs) :|: n >= 0, xs >= 0, z' = 1 + 0 + xs, z = n
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z = 0, z1 = xs
   last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   qsort(z) -{ 4 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + x + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + x + qsort(filterhigh(0, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + 0 + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(x, 1 + x + 0)), 1 + 0 + qsort(filterhigh(0, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + 0)), 1 + x + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + 0)), 1 + x + qsort(filterhigh(0, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + 0)), 1 + 0 + qsort(filterhigh(x, 1 + x + 0))) :|: x >= 0, z = 1 + x + 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { append }
   { ge }
   { last }
   { filterlow, if1 }
   { filterhigh, if2 }
   { qsort }

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {append}, {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}

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

(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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {append}, {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {append}, {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: ?, size: O(n^1) [1 + z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: append
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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ge
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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: ?, size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: ge
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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 2 }-> if2(ge(x2, z - 1), 1 + (z - 1), 1 + x2, xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 2 }-> if1(ge(z - 1, y2), 1 + (z - 1), 1 + y2, xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: 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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {last}, {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: ?, size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + last(1 + y' + xs') + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 3 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(last(1 + y' + xs'), 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 2 }-> append(qsort(filterlow(last(1 + y' + xs'), 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(last(1 + y'' + xs''), 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + last(1 + y'' + xs'') + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(last(1 + y1 + xs1), 1 + x + (1 + y1 + xs1)))) :|: z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 13 + 3*xs' + 3*y' }-> append(qsort(filterlow(s2, 1 + x + (1 + y' + xs'))), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s5, 1 + x + (1 + y' + xs'))), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s7, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 5 + xs' + y' }-> append(qsort(filterlow(s9, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 9 + 2*xs'' + 2*y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 5 + xs'' + y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 5 + xs1 + y1 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'' + z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 13 + 3*xs' + 3*y' }-> append(qsort(filterlow(s2, 1 + x + (1 + y' + xs'))), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s5, 1 + x + (1 + y' + xs'))), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s7, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 5 + xs' + y' }-> append(qsort(filterlow(s9, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 9 + 2*xs'' + 2*y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 5 + xs'' + y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 5 + xs1 + y1 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterlow,if1}, {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: ?, size: O(n^1) [z']
  if1: runtime: ?, size: O(n^1) [1 + z'' + z1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: filterlow
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 14 + 7*z'

Computed RUNTIME bound using CoFloCo for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 15 + 7*z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 4 + y2 }-> if1(s', 1 + (z - 1), 1 + y2, xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x', xs) :|: xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 1 }-> filterlow(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterlow(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 13 + 3*xs' + 3*y' }-> append(qsort(filterlow(s2, 1 + x + (1 + y' + xs'))), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s5, 1 + x + (1 + y' + xs'))), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 9 + 2*xs' + 2*y' }-> append(qsort(filterlow(s7, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 5 + xs' + y' }-> append(qsort(filterlow(s9, 1 + x + (1 + y' + xs'))), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 1 }-> append(qsort(filterlow(0, 1 + x + xs)), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 9 + 2*xs'' + 2*y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 5 + xs'' + y'' }-> append(qsort(filterlow(0, 1 + x + (1 + y'' + xs''))), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 5 + xs1 + y1 }-> append(qsort(filterlow(0, 1 + x + (1 + y1 + xs1))), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(0, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 2 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 3 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 4 }-> append(qsort(filterlow(z - 1, 1 + (z - 1) + 0)), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: z - 1 >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]

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

(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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 18 + 7*z }-> append(qsort(s15), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 17 + 7*z }-> append(qsort(s16), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 17 + 7*z }-> append(qsort(s17), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s18), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 41 + 7*x + 10*xs' + 10*y' }-> append(qsort(s19), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s20), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s21), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 33 + 7*x + 8*xs' + 8*y' }-> append(qsort(s22), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 17 + 7*z }-> append(qsort(s23), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s24), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 37 + 7*x + 9*xs'' + 9*y'' }-> append(qsort(s25), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 33 + 7*x + 8*xs'' + 8*y'' }-> append(qsort(s26), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s27), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 33 + 7*x + 8*xs1 + 8*y1 }-> append(qsort(s28), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 22 + 7*x + 7*xs }-> append(qsort(s29), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'' + z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 18 + 7*z }-> append(qsort(s15), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 17 + 7*z }-> append(qsort(s16), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 17 + 7*z }-> append(qsort(s17), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s18), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 41 + 7*x + 10*xs' + 10*y' }-> append(qsort(s19), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s20), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s21), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 33 + 7*x + 8*xs' + 8*y' }-> append(qsort(s22), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 17 + 7*z }-> append(qsort(s23), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s24), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 37 + 7*x + 9*xs'' + 9*y'' }-> append(qsort(s25), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 33 + 7*x + 8*xs'' + 8*y'' }-> append(qsort(s26), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s27), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 33 + 7*x + 8*xs1 + 8*y1 }-> append(qsort(s28), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 22 + 7*x + 7*xs }-> append(qsort(s29), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {filterhigh,if2}, {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: ?, size: O(n^1) [z']
  if2: runtime: ?, size: O(n^1) [1 + z'' + z1]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: filterhigh
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: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 6*z' + 2*z'*z1 + 8*z1

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 3 + z }-> if2(s'', 1 + (z - 1), 1 + x2, xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 2 }-> if2(1, 0, x, xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 2 }-> if2(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + z'' + filterhigh(z', z1) :|: z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 18 + 7*z }-> append(qsort(s15), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 17 + 7*z }-> append(qsort(s16), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 17 + 7*z }-> append(qsort(s17), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s18), 1 + 0 + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 41 + 7*x + 10*xs' + 10*y' }-> append(qsort(s19), 1 + s3 + qsort(filterhigh(s4, 1 + x + (1 + y' + xs')))) :|: s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s20), 1 + s6 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 37 + 7*x + 9*xs' + 9*y' }-> append(qsort(s21), 1 + 0 + qsort(filterhigh(s8, 1 + x + (1 + y' + xs')))) :|: s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 33 + 7*x + 8*xs' + 8*y' }-> append(qsort(s22), 1 + 0 + qsort(filterhigh(0, 1 + x + (1 + y' + xs')))) :|: s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 17 + 7*z }-> append(qsort(s23), 1 + (z - 1) + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s24), 1 + (z - 1) + qsort(filterhigh(0, 1 + (z - 1) + 0))) :|: s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 37 + 7*x + 9*xs'' + 9*y'' }-> append(qsort(s25), 1 + s10 + qsort(filterhigh(s11, 1 + x + (1 + y'' + xs'')))) :|: s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 33 + 7*x + 8*xs'' + 8*y'' }-> append(qsort(s26), 1 + s12 + qsort(filterhigh(0, 1 + x + (1 + y'' + xs'')))) :|: s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 16 + 7*z }-> append(qsort(s27), 1 + 0 + qsort(filterhigh(z - 1, 1 + (z - 1) + 0))) :|: s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 33 + 7*x + 8*xs1 + 8*y1 }-> append(qsort(s28), 1 + 0 + qsort(filterhigh(s13, 1 + x + (1 + y1 + xs1)))) :|: s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 22 + 7*x + 7*xs }-> append(qsort(s29), 1 + 0 + qsort(filterhigh(0, 1 + x + xs))) :|: s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]

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

(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:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 26 + 8*xs }-> s50 :|: s50 >= 0, s50 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s51 :|: s51 >= 0, s51 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s52 :|: s52 >= 0, s52 <= 1 + x2 + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s53 :|: s53 >= 0, s53 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s54 :|: s54 >= 0, s54 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 26 + 13*z + z^2 }-> append(qsort(s15), 1 + (z - 1) + qsort(s35)) :|: s35 >= 0, s35 <= 1 + (z - 1) + 0, s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 28 + 11*z }-> append(qsort(s16), 1 + (z - 1) + qsort(s36)) :|: s36 >= 0, s36 <= 1 + (z - 1) + 0, s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s17), 1 + 0 + qsort(s37)) :|: s37 >= 0, s37 <= 1 + (z - 1) + 0, s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 27 + 11*z }-> append(qsort(s18), 1 + 0 + qsort(s38)) :|: s38 >= 0, s38 <= 1 + (z - 1) + 0, s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 60 + 5*s4 + s4*x + s4*xs' + s4*y' + 11*x + 14*xs' + 14*y' }-> append(qsort(s19), 1 + s3 + qsort(s39)) :|: s39 >= 0, s39 <= 1 + x + (1 + y' + xs'), s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 56 + 11*x + 13*xs' + 13*y' }-> append(qsort(s20), 1 + s6 + qsort(s40)) :|: s40 >= 0, s40 <= 1 + x + (1 + y' + xs'), s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 56 + 5*s8 + s8*x + s8*xs' + s8*y' + 11*x + 13*xs' + 13*y' }-> append(qsort(s21), 1 + 0 + qsort(s41)) :|: s41 >= 0, s41 <= 1 + x + (1 + y' + xs'), s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 52 + 11*x + 12*xs' + 12*y' }-> append(qsort(s22), 1 + 0 + qsort(s42)) :|: s42 >= 0, s42 <= 1 + x + (1 + y' + xs'), s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s23), 1 + (z - 1) + qsort(s43)) :|: s43 >= 0, s43 <= 1 + (z - 1) + 0, s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 27 + 11*z }-> append(qsort(s24), 1 + (z - 1) + qsort(s44)) :|: s44 >= 0, s44 <= 1 + (z - 1) + 0, s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 56 + 5*s11 + s11*x + s11*xs'' + s11*y'' + 11*x + 13*xs'' + 13*y'' }-> append(qsort(s25), 1 + s10 + qsort(s45)) :|: s45 >= 0, s45 <= 1 + x + (1 + y'' + xs''), s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 52 + 11*x + 12*xs'' + 12*y'' }-> append(qsort(s26), 1 + s12 + qsort(s46)) :|: s46 >= 0, s46 <= 1 + x + (1 + y'' + xs''), s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 24 + 13*z + z^2 }-> append(qsort(s27), 1 + 0 + qsort(s47)) :|: s47 >= 0, s47 <= 1 + (z - 1) + 0, s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 52 + 5*s13 + s13*x + s13*xs1 + s13*y1 + 11*x + 12*xs1 + 12*y1 }-> append(qsort(s28), 1 + 0 + qsort(s48)) :|: s48 >= 0, s48 <= 1 + x + (1 + y1 + xs1), s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 37 + 11*x + 11*xs }-> append(qsort(s29), 1 + 0 + qsort(s49)) :|: s49 >= 0, s49 <= 1 + x + xs, s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 26 + 8*xs }-> s50 :|: s50 >= 0, s50 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s51 :|: s51 >= 0, s51 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s52 :|: s52 >= 0, s52 <= 1 + x2 + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s53 :|: s53 >= 0, s53 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s54 :|: s54 >= 0, s54 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 26 + 13*z + z^2 }-> append(qsort(s15), 1 + (z - 1) + qsort(s35)) :|: s35 >= 0, s35 <= 1 + (z - 1) + 0, s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 28 + 11*z }-> append(qsort(s16), 1 + (z - 1) + qsort(s36)) :|: s36 >= 0, s36 <= 1 + (z - 1) + 0, s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s17), 1 + 0 + qsort(s37)) :|: s37 >= 0, s37 <= 1 + (z - 1) + 0, s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 27 + 11*z }-> append(qsort(s18), 1 + 0 + qsort(s38)) :|: s38 >= 0, s38 <= 1 + (z - 1) + 0, s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 60 + 5*s4 + s4*x + s4*xs' + s4*y' + 11*x + 14*xs' + 14*y' }-> append(qsort(s19), 1 + s3 + qsort(s39)) :|: s39 >= 0, s39 <= 1 + x + (1 + y' + xs'), s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 56 + 11*x + 13*xs' + 13*y' }-> append(qsort(s20), 1 + s6 + qsort(s40)) :|: s40 >= 0, s40 <= 1 + x + (1 + y' + xs'), s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 56 + 5*s8 + s8*x + s8*xs' + s8*y' + 11*x + 13*xs' + 13*y' }-> append(qsort(s21), 1 + 0 + qsort(s41)) :|: s41 >= 0, s41 <= 1 + x + (1 + y' + xs'), s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 52 + 11*x + 12*xs' + 12*y' }-> append(qsort(s22), 1 + 0 + qsort(s42)) :|: s42 >= 0, s42 <= 1 + x + (1 + y' + xs'), s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s23), 1 + (z - 1) + qsort(s43)) :|: s43 >= 0, s43 <= 1 + (z - 1) + 0, s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 27 + 11*z }-> append(qsort(s24), 1 + (z - 1) + qsort(s44)) :|: s44 >= 0, s44 <= 1 + (z - 1) + 0, s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 56 + 5*s11 + s11*x + s11*xs'' + s11*y'' + 11*x + 13*xs'' + 13*y'' }-> append(qsort(s25), 1 + s10 + qsort(s45)) :|: s45 >= 0, s45 <= 1 + x + (1 + y'' + xs''), s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 52 + 11*x + 12*xs'' + 12*y'' }-> append(qsort(s26), 1 + s12 + qsort(s46)) :|: s46 >= 0, s46 <= 1 + x + (1 + y'' + xs''), s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 24 + 13*z + z^2 }-> append(qsort(s27), 1 + 0 + qsort(s47)) :|: s47 >= 0, s47 <= 1 + (z - 1) + 0, s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 52 + 5*s13 + s13*x + s13*xs1 + s13*y1 + 11*x + 12*xs1 + 12*y1 }-> append(qsort(s28), 1 + 0 + qsort(s48)) :|: s48 >= 0, s48 <= 1 + x + (1 + y1 + xs1), s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 37 + 11*x + 11*xs }-> append(qsort(s29), 1 + 0 + qsort(s49)) :|: s49 >= 0, s49 <= 1 + x + xs, s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]
  qsort: runtime: ?, size: INF

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   append(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 26 + 8*xs }-> s50 :|: s50 >= 0, s50 <= x + xs + 1, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   filterhigh(z, z') -{ 18 + 4*z + 2*z*z' + 8*z' }-> s51 :|: s51 >= 0, s51 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0
   filterhigh(z, z') -{ 27 + 8*xs + 2*xs*z + 7*z }-> s52 :|: s52 >= 0, s52 <= 1 + x2 + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + x2) + xs, z - 1 >= 0, x2 >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   filterlow(z, z') -{ 10 + 7*z' }-> s30 :|: s30 >= 0, s30 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   filterlow(z, z') -{ 17 + 7*xs }-> s31 :|: s31 >= 0, s31 <= 1 + x' + xs + 1, xs >= 0, z' = 1 + (1 + x') + xs, x' >= 0, z = 0
   filterlow(z, z') -{ 19 + 7*xs + y2 }-> s32 :|: s32 >= 0, s32 <= 1 + y2 + xs + 1, s' >= 0, s' <= 1, xs >= 0, z' = 1 + (1 + y2) + xs, z - 1 >= 0, y2 >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s33 :|: s33 >= 0, s33 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s34 :|: s34 >= 0, s34 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> s53 :|: s53 >= 0, s53 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0
   if2(z, z', z'', z1) -{ 12 + 3*z' + z'*z1 + 4*z1 }-> 1 + z'' + s54 :|: s54 >= 0, s54 <= z1, z' >= 0, z1 >= 0, z'' >= 0, z = 0
   last(z) -{ 4 + xs + y }-> s14 :|: s14 >= 0, s14 <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: z >= 0
   last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   qsort(z) -{ 26 + 13*z + z^2 }-> append(qsort(s15), 1 + (z - 1) + qsort(s35)) :|: s35 >= 0, s35 <= 1 + (z - 1) + 0, s15 >= 0, s15 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 28 + 11*z }-> append(qsort(s16), 1 + (z - 1) + qsort(s36)) :|: s36 >= 0, s36 <= 1 + (z - 1) + 0, s16 >= 0, s16 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s17), 1 + 0 + qsort(s37)) :|: s37 >= 0, s37 <= 1 + (z - 1) + 0, s17 >= 0, s17 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 27 + 11*z }-> append(qsort(s18), 1 + 0 + qsort(s38)) :|: s38 >= 0, s38 <= 1 + (z - 1) + 0, s18 >= 0, s18 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 60 + 5*s4 + s4*x + s4*xs' + s4*y' + 11*x + 14*xs' + 14*y' }-> append(qsort(s19), 1 + s3 + qsort(s39)) :|: s39 >= 0, s39 <= 1 + x + (1 + y' + xs'), s19 >= 0, s19 <= 1 + x + (1 + y' + xs'), s2 >= 0, s2 <= 1 + y' + xs', s3 >= 0, s3 <= 1 + y' + xs', s4 >= 0, s4 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 56 + 11*x + 13*xs' + 13*y' }-> append(qsort(s20), 1 + s6 + qsort(s40)) :|: s40 >= 0, s40 <= 1 + x + (1 + y' + xs'), s20 >= 0, s20 <= 1 + x + (1 + y' + xs'), s5 >= 0, s5 <= 1 + y' + xs', s6 >= 0, s6 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 56 + 5*s8 + s8*x + s8*xs' + s8*y' + 11*x + 13*xs' + 13*y' }-> append(qsort(s21), 1 + 0 + qsort(s41)) :|: s41 >= 0, s41 <= 1 + x + (1 + y' + xs'), s21 >= 0, s21 <= 1 + x + (1 + y' + xs'), s7 >= 0, s7 <= 1 + y' + xs', s8 >= 0, s8 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 52 + 11*x + 12*xs' + 12*y' }-> append(qsort(s22), 1 + 0 + qsort(s42)) :|: s42 >= 0, s42 <= 1 + x + (1 + y' + xs'), s22 >= 0, s22 <= 1 + x + (1 + y' + xs'), s9 >= 0, s9 <= 1 + y' + xs', x >= 0, xs' >= 0, y' >= 0, z = 1 + x + (1 + y' + xs')
   qsort(z) -{ 25 + 13*z + z^2 }-> append(qsort(s23), 1 + (z - 1) + qsort(s43)) :|: s43 >= 0, s43 <= 1 + (z - 1) + 0, s23 >= 0, s23 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 27 + 11*z }-> append(qsort(s24), 1 + (z - 1) + qsort(s44)) :|: s44 >= 0, s44 <= 1 + (z - 1) + 0, s24 >= 0, s24 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 56 + 5*s11 + s11*x + s11*xs'' + s11*y'' + 11*x + 13*xs'' + 13*y'' }-> append(qsort(s25), 1 + s10 + qsort(s45)) :|: s45 >= 0, s45 <= 1 + x + (1 + y'' + xs''), s25 >= 0, s25 <= 1 + x + (1 + y'' + xs''), s10 >= 0, s10 <= 1 + y'' + xs'', s11 >= 0, s11 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 52 + 11*x + 12*xs'' + 12*y'' }-> append(qsort(s26), 1 + s12 + qsort(s46)) :|: s46 >= 0, s46 <= 1 + x + (1 + y'' + xs''), s26 >= 0, s26 <= 1 + x + (1 + y'' + xs''), s12 >= 0, s12 <= 1 + y'' + xs'', z = 1 + x + (1 + y'' + xs''), x >= 0, xs'' >= 0, y'' >= 0
   qsort(z) -{ 24 + 13*z + z^2 }-> append(qsort(s27), 1 + 0 + qsort(s47)) :|: s47 >= 0, s47 <= 1 + (z - 1) + 0, s27 >= 0, s27 <= 1 + (z - 1) + 0, z - 1 >= 0
   qsort(z) -{ 52 + 5*s13 + s13*x + s13*xs1 + s13*y1 + 11*x + 12*xs1 + 12*y1 }-> append(qsort(s28), 1 + 0 + qsort(s48)) :|: s48 >= 0, s48 <= 1 + x + (1 + y1 + xs1), s28 >= 0, s28 <= 1 + x + (1 + y1 + xs1), s13 >= 0, s13 <= 1 + y1 + xs1, z = 1 + x + (1 + y1 + xs1), y1 >= 0, x >= 0, xs1 >= 0
   qsort(z) -{ 37 + 11*x + 11*xs }-> append(qsort(s29), 1 + 0 + qsort(s49)) :|: s49 >= 0, s49 <= 1 + x + xs, s29 >= 0, s29 <= 1 + x + xs, z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {qsort}
Previous analysis results are:
  append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  last: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  filterlow: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [z']
  if1: runtime: O(n^1) [15 + 7*z1], size: O(n^1) [1 + z'' + z1]
  filterhigh: runtime: O(n^2) [11 + 3*z + z*z' + 4*z'], size: O(n^1) [z']
  if2: runtime: O(n^2) [24 + 6*z' + 2*z'*z1 + 8*z1], size: O(n^1) [1 + z'' + z1]
  qsort: runtime: INF, size: INF

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

(55) 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:

   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]
   last(v0) -> null_last [0]
   ge(v0, v1) -> null_ge [0]
   if1(v0, v1, v2, v3) -> null_if1 [0]
   if2(v0, v1, v2, v3) -> null_if2 [0]

And the following fresh constants:    null_qsort, null_filterlow, null_filterhigh, null_append, null_last, null_ge, null_if1, null_if2

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

(56)
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:

   qsort(nil) -> nil [1]
   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) [1]
   filterlow(n, nil) -> nil [1]
   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1]
   if1(true, n, x, xs) -> filterlow(n, xs) [1]
   if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1]
   filterhigh(n, nil) -> nil [1]
   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1]
   if2(true, n, x, xs) -> filterhigh(n, xs) [1]
   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   append(nil, ys) -> ys [1]
   append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1]
   last(nil) -> 0 [1]
   last(cons(x, nil)) -> x [1]
   last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [1]
   qsort(v0) -> null_qsort [0]
   filterlow(v0, v1) -> null_filterlow [0]
   filterhigh(v0, v1) -> null_filterhigh [0]
   append(v0, v1) -> null_append [0]
   last(v0) -> null_last [0]
   ge(v0, v1) -> null_ge [0]
   if1(v0, v1, v2, v3) -> null_if1 [0]
   if2(v0, v1, v2, v3) -> null_if2 [0]

The TRS has the following type information:
qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
nil :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
cons :: 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
filterlow :: 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
last :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> 0:s:null_last
filterhigh :: 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
if1 :: true:false:null_ge -> 0:s:null_last -> 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
ge :: 0:s:null_last -> 0:s:null_last -> true:false:null_ge
true :: true:false:null_ge
false :: true:false:null_ge
if2 :: true:false:null_ge -> 0:s:null_last -> 0:s:null_last -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
0 :: 0:s:null_last
s :: 0:s:null_last -> 0:s:null_last
ys :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
null_qsort :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
null_filterlow :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
null_filterhigh :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
null_append :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
null_last :: 0:s:null_last
null_ge :: true:false:null_ge
null_if1 :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2
null_if2 :: nil:cons:ys:null_qsort:null_filterlow:null_filterhigh:null_append:null_if1:null_if2

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

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

   nil => 0
   true => 2
   false => 1
   0 => 0
   ys => 1
   null_qsort => 0
   null_filterlow => 0
   null_filterhigh => 0
   null_append => 0
   null_last => 0
   null_ge => 0
   null_if1 => 0
   null_if2 => 0

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

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

   append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0
   append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1
   filterhigh(z, z') -{ 1 }-> if2(ge(x, n), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0
   filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   filterlow(z, z') -{ 1 }-> if1(ge(n, x), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0
   filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0
   filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: z = 2, n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z1 = xs
   if1(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: z = 2, n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z1 = xs
   if2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs
   last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0
   last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs)
   last(z) -{ 1 }-> 0 :|: z = 0
   last(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   qsort(z) -{ 1 }-> append(qsort(filterlow(last(1 + x + xs), 1 + x + xs)), 1 + last(1 + x + xs) + qsort(filterhigh(last(1 + x + xs), 1 + x + xs))) :|: z = 1 + x + xs, xs >= 0, x >= 0
   qsort(z) -{ 1 }-> 0 :|: z = 0
   qsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1)))))
   filterlow(z0, nil) -> nil
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, nil) -> nil
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
   last(nil) -> 0
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
Tuples:
   QSORT(nil) -> c
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(z0, nil) -> c4
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, nil) -> c8
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(z0, 0) -> c12
   GE(0, s(z0)) -> c13
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(nil, ys) -> c15
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(nil) -> c17
   LAST(cons(z0, nil)) -> c18
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
S tuples:
   QSORT(nil) -> c
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(z0, nil) -> c4
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, nil) -> c8
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(z0, 0) -> c12
   GE(0, s(z0)) -> c13
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(nil, ys) -> c15
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(nil) -> c17
   LAST(cons(z0, nil)) -> c18
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, last_1

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1

Compound Symbols:   c, c1_4, c2_2, c3_4, c4, c5_2, c6_1, c7_1, c8, c9_2, c10_1, c11_1, c12, c13, c14_1, c15, c16_1, c17, c18, c19_1


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   GE(z0, 0) -> c12
   FILTERHIGH(z0, nil) -> c8
   FILTERLOW(z0, nil) -> c4
   APPEND(nil, ys) -> c15
   QSORT(nil) -> c
   LAST(cons(z0, nil)) -> c18
   LAST(nil) -> c17
   GE(0, s(z0)) -> c13

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1)))))
   filterlow(z0, nil) -> nil
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, nil) -> nil
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
   last(nil) -> 0
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
Tuples:
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
S tuples:
   QSORT(cons(z0, z1)) -> c1(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(APPEND(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1))))), QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, last_1

Defined Pair Symbols:   QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1

Compound Symbols:   c1_4, c2_2, c3_4, c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1


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

(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1)))))
   filterlow(z0, nil) -> nil
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, nil) -> nil
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
   last(nil) -> 0
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   qsort_1, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, last_1

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1

Compound Symbols:   c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3


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

(65) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   qsort(nil) -> nil
   qsort(cons(z0, z1)) -> append(qsort(filterlow(last(cons(z0, z1)), cons(z0, z1))), cons(last(cons(z0, z1)), qsort(filterhigh(last(cons(z0, z1)), cons(z0, z1)))))
   append(nil, ys) -> ys
   append(cons(z0, z1), ys) -> cons(z0, append(z1, ys))
   last(nil) -> 0

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1

Compound Symbols:   c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3


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

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = x_1
   POL(QSORT(x_1)) = [1]
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1] + x_1
   POL(ge(x_1, x_2)) = x_1 + x_2
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(last(x_1)) = 0
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]
   POL(ys) = 0

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1

Compound Symbols:   c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
We considered the (Usable) Rules:none
And the Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(APPEND(x_1, x_2)) = x_1
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = 0
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = 0
   POL(QSORT(x_1)) = 0
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1]
   POL(ge(x_1, x_2)) = x_1 + x_2
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(last(x_1)) = 0
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = [1]
   POL(ys) = 0

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
S tuples:
   FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1

Compound Symbols:   c5_2, c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERLOW(z0, cons(z1, z2)) -> c5(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by 
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2), GE(z0, 0))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2), GE(0, s(z0)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2), GE(z0, 0))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2), GE(0, s(z0)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2), GE(z0, 0))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2), GE(0, s(z0)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2

Compound Symbols:   c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2

Compound Symbols:   c6_1, c7_1, c9_2, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2, c5_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERHIGH(z0, cons(z1, z2)) -> c9(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) by 
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2), GE(z0, 0))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2), GE(0, s(z0)))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2), GE(z0, 0))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2), GE(0, s(z0)))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2), GE(z0, 0))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2), GE(0, s(z0)))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2, c5_1, c9_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c1_3, c2_1, c3_3, c5_2, c5_1, c9_2, c9_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, z1)) -> c1(QSORT(filterlow(last(cons(z0, z1)), cons(z0, z1))), FILTERLOW(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) by 
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c3_3, c5_2, c5_1, c9_2, c9_1, c1_3


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

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c3_3, c5_2, c5_1, c9_2, c9_1, c1_3, c1_2


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, z1)) -> c3(QSORT(filterhigh(last(cons(z0, z1)), cons(z0, z1))), FILTERHIGH(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) by 
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_2, c5_1, c9_2, c9_1, c1_3, c1_2, c3_3


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_2, c5_1, c9_2, c9_1, c1_3, c1_2, c3_3, c3_2


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERLOW(s(z0), cons(s(z1), x2)) -> c5(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by 
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1)))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_2, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FILTERHIGH(s(z1), cons(s(z0), x2)) -> c9(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) by 
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z1, z2)) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), FILTERLOW(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) by 
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2


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

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2


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

(95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterhigh(z0, nil) -> nil
   filterlow(z0, nil) -> nil
   ge(0, s(z0)) -> false
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   ge(z0, 0) -> true
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
And the Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = x_1
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = 0
   POL(QSORT(x_1)) = [1] + x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(false) = [1]
   POL(filterhigh(x_1, x_2)) = x_2
   POL(filterlow(x_1, x_2)) = x_2
   POL(ge(x_1, x_2)) = [1]
   POL(if1(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(last(x_1)) = 0
   POL(nil) = [1]
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_3, c1_2, c3_3, c3_2, c5_2, c9_2


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by 
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c1(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c1(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c1(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_2, c3_3, c3_2, c5_2, c9_2, c1_3


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

(99) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_2, c3_3, c3_2, c5_2, c9_2, c1_3, c_1


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

(101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
We considered the (Usable) Rules:none
And the Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = x_1
   POL(QSORT(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1]
   POL(ge(x_1, x_2)) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(last(x_1)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c1_2, c3_3, c3_2, c5_2, c9_2, c1_3, c_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, nil)) -> c1(QSORT(filterlow(z0, cons(z0, nil))), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by 
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1


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

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
We considered the (Usable) Rules:none
And the Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = x_1
   POL(QSORT(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1]
   POL(ge(x_1, x_2)) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(last(x_1)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z1, z2)) -> c3(QSORT(if2(ge(z1, last(cons(z1, z2))), last(cons(z1, z2)), z1, z2)), FILTERHIGH(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) by 
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1


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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_3, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by 
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c3(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c3(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c3(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3


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

(113) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c4_1


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

(115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
We considered the (Usable) Rules:none
And the Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = x_1
   POL(QSORT(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1]
   POL(ge(x_1, x_2)) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(last(x_1)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c3_2, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c4_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QSORT(cons(z0, nil)) -> c3(QSORT(filterhigh(z0, cons(z0, nil))), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by 
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1


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

(119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
We considered the (Usable) Rules:none
And the Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APPEND(x_1, x_2)) = 0
   POL(FILTERHIGH(x_1, x_2)) = 0
   POL(FILTERLOW(x_1, x_2)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(IF1(x_1, x_2, x_3, x_4)) = x_3
   POL(IF2(x_1, x_2, x_3, x_4)) = 0
   POL(LAST(x_1)) = x_1
   POL(QSORT(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(false) = 0
   POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2
   POL(filterlow(x_1, x_2)) = [1]
   POL(ge(x_1, x_2)) = 0
   POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(last(x_1)) = 0
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = 0
   POL(ys) = 0

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1))) by QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c1_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2))
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c6_1, c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_2, c1_3, c_1, c1_1, c3_3, c3_2, c4_1, c3_1


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

(127) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(true, z0, z1, z2) -> c6(FILTERLOW(z0, z2)) by 
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
S tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_2, c1_3, c_1, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
S tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_2, c1_3, c_1, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
S tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z1, cons(x1, x2))) -> c1(QSORT(if1(ge(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERLOW(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2))
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF1_4, IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2

Compound Symbols:   c7_1, c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1


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

(135) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF1(false, z0, z1, z2) -> c7(FILTERLOW(z0, z2)) by 
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
S tuples:
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4

Compound Symbols:   c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1, c7_1


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

(137) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, cons(z0, nil))) -> c1(QSORT(filterlow(z0, cons(x0, cons(z0, nil)))), FILTERLOW(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) by QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
S tuples:
   IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2))
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4

Compound Symbols:   c10_1, c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1, c7_1


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

(139) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(true, z0, z1, z2) -> c10(FILTERHIGH(z0, z2)) by 
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
S tuples:
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4

Compound Symbols:   c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c1_3, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c6_1, c7_1, c10_1


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c1(QSORT(filterlow(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERLOW(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) by QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
S tuples:
   IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2))
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   IF2_4, GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4

Compound Symbols:   c11_1, c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1


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

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF2(false, z0, z1, z2) -> c11(FILTERHIGH(z0, z2)) by 
   IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1)))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1)))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
S tuples:
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1)))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(0, x1)) -> c1(QSORT(if1(true, last(cons(0, x1)), 0, x1)), FILTERLOW(last(cons(0, x1)), cons(0, x1)), LAST(cons(0, x1)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4

Compound Symbols:   c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1


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

(145) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF2(false, 0, z1, cons(y0, y1)) -> c11(FILTERHIGH(0, cons(y0, y1)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
S tuples:
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4

Compound Symbols:   c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
S tuples:
   GE(s(z0), s(z1)) -> c14(GE(z0, z1))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   GE_2, APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4

Compound Symbols:   c14_1, c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1


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

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(z0), s(z1)) -> c14(GE(z0, z1)) by 
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0)))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0))))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1


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

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_2, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c1_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil))) by QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c3_2, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_2, c3_3, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_2, c3_3, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z1, cons(x1, x2))) -> c3(QSORT(if2(ge(z1, last(cons(x1, x2))), last(cons(x1, x2)), z1, cons(x1, x2))), FILTERHIGH(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(167) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, cons(z0, nil))) -> c3(QSORT(filterhigh(z0, cons(x0, cons(z0, nil)))), FILTERHIGH(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) by QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c3_3, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, cons(z0, cons(z1, z2)))) -> c3(QSORT(filterhigh(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), FILTERHIGH(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) by QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3


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

(171) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c3_2, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3


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

(173) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(last(cons(z0, nil)), cons(z0, nil))) by QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c3_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2


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

(175) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil))) by QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1)))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1


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

(177) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(false, s(y0), z1, cons(0, y1)) -> c11(FILTERHIGH(s(y0), cons(0, y1))) by 
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1


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

(181) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1


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

(183) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(last(cons(z0, nil)), z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c1_2, c7_1, c10_1, c11_1, c14_1, c1_1, c3_3, c3_2, c3_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERLOW(z0, cons(z0, nil))) by QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1


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

(187) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(false, s(0), z1, cons(s(y0), y1)) -> c11(FILTERHIGH(s(0), cons(s(y0), y1))) by 
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1)))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1


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

(189) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(false, s(s(y0)), z1, cons(s(0), y1)) -> c11(FILTERHIGH(s(s(y0)), cons(s(0), y1))) by 
   IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))
   IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2)))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))
   IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1


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

(191) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF2(false, s(s(y0)), z1, cons(s(s(y1)), y2)) -> c11(FILTERHIGH(s(s(y0)), cons(s(s(y1)), y2))) by 
   IF2(false, s(s(z0)), 0, cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(z0)), cons(s(s(z2)), z3)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3)))
   IF2(false, s(s(x0)), s(0), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3)))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2)
   filterlow(z0, nil) -> nil
   last(cons(z0, nil)) -> z0
   last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2))
   if1(true, z0, z1, z2) -> filterlow(z0, z2)
   if1(false, z0, z1, z2) -> cons(z1, filterlow(z0, z2))
   filterhigh(z0, cons(z1, z2)) -> if2(ge(z1, z0), z0, z1, z2)
   filterhigh(z0, nil) -> nil
   if2(true, z0, z1, z2) -> filterhigh(z0, z2)
   if2(false, z0, z1, z2) -> cons(z1, filterhigh(z0, z2))
Tuples:
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(0, z0)) -> c1(QSORT(filterlow(last(cons(0, z0)), z0)), FILTERLOW(last(cons(0, z0)), cons(0, z0)), LAST(cons(0, z0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   QSORT(cons(z0, cons(z1, z2))) -> c(FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c1(FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, cons(z1, z2))) -> c4(FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))
   IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(z0)), 0, cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(z0)), cons(s(s(z2)), z3)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3)))
   IF2(false, s(s(x0)), s(0), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3)))
S tuples:
   LAST(cons(z0, cons(z1, z2))) -> c19(LAST(cons(z1, z2)))
   FILTERLOW(z0, cons(0, x2)) -> c5(IF1(true, z0, 0, x2))
   FILTERLOW(0, cons(s(z0), x2)) -> c5(IF1(false, 0, s(z0), x2))
   FILTERHIGH(0, cons(z0, x2)) -> c9(IF2(true, 0, z0, x2))
   FILTERHIGH(s(z0), cons(0, x2)) -> c9(IF2(false, s(z0), 0, x2))
   FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c5(IF1(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERLOW(s(x0), cons(s(x1), x2)) -> c5(GE(s(x0), s(x1)))
   FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c9(IF2(ge(z0, z1), s(s(z1)), s(s(z0)), x2), GE(s(s(z0)), s(s(z1))))
   FILTERHIGH(s(x0), cons(s(x1), x2)) -> c9(GE(s(x1), s(x0)))
   IF1(true, x0, 0, x1) -> c6(FILTERLOW(x0, x1))
   IF1(true, s(x0), s(0), x1) -> c6(FILTERLOW(s(x0), x1))
   IF1(true, s(s(x0)), s(s(x1)), x2) -> c6(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, z2))) -> c1(QSORT(if1(ge(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERLOW(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   IF1(false, 0, s(x0), x1) -> c7(FILTERLOW(0, x1))
   IF1(false, s(0), s(s(x0)), x1) -> c7(FILTERLOW(s(0), x1))
   IF1(false, s(s(x0)), s(s(x1)), x2) -> c7(FILTERLOW(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, nil))) -> c1(QSORT(filterlow(z1, cons(z0, cons(z1, nil)))), FILTERLOW(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   IF2(true, 0, x0, x1) -> c10(FILTERHIGH(0, x1))
   IF2(true, s(0), s(x0), x1) -> c10(FILTERHIGH(s(0), x1))
   IF2(true, s(s(x0)), s(s(x1)), x2) -> c10(FILTERHIGH(s(s(x0)), x2))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c1(QSORT(filterlow(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERLOW(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   IF2(false, s(y0), z1, cons(s(y1), y2)) -> c11(FILTERHIGH(s(y0), cons(s(y1), y2)))
   GE(s(s(y0)), s(s(y1))) -> c14(GE(s(y0), s(y1)))
   FILTERLOW(s(z0), cons(s(0), x2)) -> c5(IF1(true, s(z0), s(0), x2))
   FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c5(IF1(false, s(0), s(s(z0)), x2))
   FILTERHIGH(s(0), cons(s(z0), x2)) -> c9(IF2(true, s(0), s(z0), x2))
   FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c9(IF2(false, s(s(z0)), s(0), x2))
   QSORT(cons(z0, nil)) -> c1(QSORT(if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z0, cons(z1, z2)))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, last(cons(z0, nil))), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), last(cons(z0, nil)), z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   QSORT(cons(z0, cons(z1, z2))) -> c3(QSORT(if2(ge(z0, last(cons(z1, z2))), last(cons(z1, z2)), z0, cons(z1, z2))), FILTERHIGH(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2))))
   QSORT(cons(z0, cons(z1, nil))) -> c3(QSORT(filterhigh(z1, cons(z0, cons(z1, nil)))), FILTERHIGH(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil))))
   QSORT(cons(z0, cons(z1, cons(z2, z3)))) -> c3(QSORT(filterhigh(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), FILTERHIGH(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3)))))
   QSORT(cons(z0, nil)) -> c3(QSORT(if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(z0, cons(z0, nil)))
   IF2(false, s(x0), 0, cons(0, z2)) -> c11(FILTERHIGH(s(x0), cons(0, z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(s(x0)), s(0), cons(0, z2)) -> c11(FILTERHIGH(s(s(x0)), cons(0, z2)))
   IF2(false, s(0), 0, cons(s(z1), z2)) -> c11(FILTERHIGH(s(0), cons(s(z1), z2)))
   IF2(false, s(s(z0)), 0, cons(s(0), z2)) -> c11(FILTERHIGH(s(s(z0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(x0)), s(0), cons(s(0), z2)) -> c11(FILTERHIGH(s(s(x0)), cons(s(0), z2)))
   IF2(false, s(s(z0)), 0, cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(z0)), cons(s(s(z2)), z3)))
   IF2(false, s(s(x0)), s(s(x1)), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3)))
   IF2(false, s(s(x0)), s(0), cons(s(s(z2)), z3)) -> c11(FILTERHIGH(s(s(x0)), cons(s(s(z2)), z3)))
K tuples:
   QSORT(cons(z0, z1)) -> c2(LAST(cons(z0, z1)))
   APPEND(cons(z0, z1), ys) -> c16(APPEND(z1, ys))
   QSORT(cons(x0, cons(x1, x2))) -> c(FILTERLOW(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c1(FILTERLOW(last(cons(x0, nil)), cons(x0, nil)))
   QSORT(cons(x0, cons(x1, x2))) -> c4(FILTERHIGH(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))))
   QSORT(cons(x0, cons(x1, x2))) -> c4(LAST(cons(x0, cons(x1, x2))))
   QSORT(cons(x0, nil)) -> c3(FILTERHIGH(last(cons(x0, nil)), cons(x0, nil)))
Defined Rule Symbols:   ge_2, filterlow_2, last_1, if1_4, filterhigh_2, if2_4

Defined Pair Symbols:   APPEND_2, LAST_1, QSORT_1, FILTERLOW_2, FILTERHIGH_2, IF1_4, IF2_4, GE_2

Compound Symbols:   c16_1, c19_1, c2_1, c5_1, c9_1, c5_2, c9_2, c_1, c4_1, c1_3, c6_1, c7_1, c10_1, c11_1, c14_1, c1_2, c1_1, c3_3, c3_2, c3_1