KILLED
proof of input_eVPq8ySgJC.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 216 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(8) CpxWeightedTrs
    (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 188 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 9 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 348 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 316 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 323 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
        (46) CpxRNTS
        (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 914 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 267 ms]
        (52) CpxRNTS
        (53) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 2184 ms]
        (56) CpxRNTS
        (57) IntTrsBoundProof [UPPER BOUND(ID), 1009 ms]
        (58) CpxRNTS
        (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 4233 ms]
        (62) CpxRNTS
        (63) IntTrsBoundProof [UPPER BOUND(ID), 568 ms]
        (64) CpxRNTS
    (65) CompletionProof [UPPER BOUND(ID), 0 ms]
    (66) CpxTypedWeightedCompleteTrs
        (67) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms]
        (68) CpxRNTS
(69) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(70) CdtProblem
    (71) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (72) CdtProblem
    (73) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 218 ms]
    (76) CdtProblem
    (77) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 94 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 84 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 81 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 93 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) 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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (206) CdtProblem
    (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem


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

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


The TRS R consists of the following rules:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
   partLt(x, Nil) -> Nil
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
   partGt(x, Nil) -> Nil
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
   goal(xs) -> quicksort(xs)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   >(S(x), S(y)) -> >(x, y)
   >(0, y) -> False
   >(S(x), 0) -> True
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
   partLt(x, Nil) -> Nil
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
   partGt(x, Nil) -> Nil
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
   goal(xs) -> quicksort(xs)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   >(S(x), S(y)) -> >(x, y)
   >(0, y) -> False
   >(S(x), 0) -> True
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
   partLt(x, Nil) -> Nil
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
   partGt(x, Nil) -> Nil
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
   goal(xs) -> quicksort(xs)

The (relative) TRS S consists of the following rules:

   <(S(x), S(y)) -> <(x, y)
   <(0', S(y)) -> True
   <(x, 0') -> False
   >(S(x), S(y)) -> >(x, y)
   >(0', y) -> False
   >(S(x), 0') -> True
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

(6)
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:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs))
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
   partLt(x, Nil) -> Nil
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
   partGt(x, Nil) -> Nil
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
   goal(xs) -> quicksort(xs)
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   >(S(x), S(y)) -> >(x, y)
   >(0, y) -> False
   >(S(x), 0) -> True
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs))
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs))
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs)
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs)

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

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

(8)
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:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs)) [1]
   partLt(x, Nil) -> Nil [1]
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs)) [1]
   partGt(x, Nil) -> Nil [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   app(Nil, ys) -> ys [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1]
   goal(xs) -> quicksort(xs) [1]
   <(S(x), S(y)) -> <(x, y) [0]
   <(0, S(y)) -> True [0]
   <(x, 0) -> False [0]
   >(S(x), S(y)) -> >(x, y) [0]
   >(0, y) -> False [0]
   >(S(x), 0) -> True [0]
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0]
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0]
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0]
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0]

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

(9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

< => lt
> => gr

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

(10)
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:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1]
   partLt(x, Nil) -> Nil [1]
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1]
   partGt(x, Nil) -> Nil [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   app(Nil, ys) -> ys [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1]
   goal(xs) -> quicksort(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   gr(S(x), S(y)) -> gr(x, y) [0]
   gr(0, y) -> False [0]
   gr(S(x), 0) -> True [0]
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0]
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0]
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0]
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1]
   partLt(x, Nil) -> Nil [1]
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1]
   partGt(x, Nil) -> Nil [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   app(Nil, ys) -> ys [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1]
   goal(xs) -> quicksort(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   gr(S(x), S(y)) -> gr(x, y) [0]
   gr(0, y) -> False [0]
   gr(S(x), 0) -> True [0]
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0]
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0]
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0]
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0]

The TRS has the following type information:
quicksort :: Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
part :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
partLt :: S:0 -> Cons:Nil -> Cons:Nil
partLt[Ite][True][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False
partGt :: S:0 -> Cons:Nil -> Cons:Nil
partGt[Ite][True][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil
gr :: S:0 -> S:0 -> True:False
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
0 :: S:0

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

(13) 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:

   notEmpty_1
   goal_1

(c) The following functions are completely defined:

   quicksort_1
   partLt_2
   partGt_2
   part_2
   app_2
   lt_2
   gr_2
   partLt[Ite][True][Ite]_3
   partGt[Ite][True][Ite]_3

Due to the following rules being added:

   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   partLt[Ite][True][Ite](v0, v1, v2) -> Nil [0]
   partGt[Ite][True][Ite](v0, v1, v2) -> Nil [0]

And the following fresh constants:    null_lt, null_gr

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

(14)
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:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1]
   partLt(x, Nil) -> Nil [1]
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1]
   partGt(x, Nil) -> Nil [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   app(Nil, ys) -> ys [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1]
   goal(xs) -> quicksort(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   gr(S(x), S(y)) -> gr(x, y) [0]
   gr(0, y) -> False [0]
   gr(S(x), 0) -> True [0]
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0]
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0]
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0]
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0]
   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   partLt[Ite][True][Ite](v0, v1, v2) -> Nil [0]
   partGt[Ite][True][Ite](v0, v1, v2) -> Nil [0]

The TRS has the following type information:
quicksort :: Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
part :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
partLt :: S:0 -> Cons:Nil -> Cons:Nil
partLt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False:null_lt:null_gr
partGt :: S:0 -> Cons:Nil -> Cons:Nil
partGt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil
gr :: S:0 -> S:0 -> True:False:null_lt:null_gr
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False:null_lt:null_gr
True :: True:False:null_lt:null_gr
False :: True:False:null_lt:null_gr
goal :: Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
0 :: S:0
null_lt :: True:False:null_lt:null_gr
null_gr :: True:False:null_lt:null_gr

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

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

(16)
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:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   partLt(S(y'), Cons(S(x''), xs)) -> partLt[Ite][True][Ite](lt(x'', y'), S(y'), Cons(S(x''), xs)) [1]
   partLt(S(y''), Cons(0, xs)) -> partLt[Ite][True][Ite](True, S(y''), Cons(0, xs)) [1]
   partLt(0, Cons(x, xs)) -> partLt[Ite][True][Ite](False, 0, Cons(x, xs)) [1]
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](null_lt, x', Cons(x, xs)) [1]
   partLt(x, Nil) -> Nil [1]
   partGt(S(y1), Cons(S(x1), xs)) -> partGt[Ite][True][Ite](gr(x1, y1), S(y1), Cons(S(x1), xs)) [1]
   partGt(x', Cons(0, xs)) -> partGt[Ite][True][Ite](False, x', Cons(0, xs)) [1]
   partGt(0, Cons(S(x2), xs)) -> partGt[Ite][True][Ite](True, 0, Cons(S(x2), xs)) [1]
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](null_gr, x', Cons(x, xs)) [1]
   partGt(x, Nil) -> Nil [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   app(Nil, ys) -> ys [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   part(x, Cons(x3, xs')) -> app(quicksort(partLt[Ite][True][Ite](lt(x3, x), x, Cons(x3, xs'))), Cons(x, quicksort(partGt[Ite][True][Ite](gr(x3, x), x, Cons(x3, xs'))))) [3]
   part(x, Nil) -> app(quicksort(Nil), Cons(x, quicksort(Nil))) [3]
   goal(xs) -> quicksort(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   gr(S(x), S(y)) -> gr(x, y) [0]
   gr(0, y) -> False [0]
   gr(S(x), 0) -> True [0]
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0]
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0]
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0]
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0]
   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   partLt[Ite][True][Ite](v0, v1, v2) -> Nil [0]
   partGt[Ite][True][Ite](v0, v1, v2) -> Nil [0]

The TRS has the following type information:
quicksort :: Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
part :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
partLt :: S:0 -> Cons:Nil -> Cons:Nil
partLt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False:null_lt:null_gr
partGt :: S:0 -> Cons:Nil -> Cons:Nil
partGt[Ite][True][Ite] :: True:False:null_lt:null_gr -> S:0 -> Cons:Nil -> Cons:Nil
gr :: S:0 -> S:0 -> True:False:null_lt:null_gr
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False:null_lt:null_gr
True :: True:False:null_lt:null_gr
False :: True:False:null_lt:null_gr
goal :: Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
0 :: S:0
null_lt :: True:False:null_lt:null_gr
null_gr :: True:False:null_lt:null_gr

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

(17) 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
   null_lt => 0
   null_gr => 0

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

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

   app(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(xs) :|: xs >= 0, z = xs
   gr(z, z') -{ 0 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 0 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: y >= 0, z = 0, z' = y
   gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, x), x, 1 + x3 + xs')), 1 + x + quicksort(partGt[Ite][True][Ite](gr(x3, x), x, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', x >= 0, xs' >= 0, z = x, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + x + quicksort(0)) :|: x >= 0, z = x, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, y1), 1 + y1, 1 + (1 + x1) + xs) :|: z = 1 + y1, y1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, x', 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, x' >= 0, z = x'
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   partGt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', y'), 1 + y', 1 + (1 + x'') + xs) :|: xs >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + y'', 1 + 0 + xs) :|: xs >= 0, z = 1 + y'', z' = 1 + 0 + xs, y'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   partLt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 0


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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0


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

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

   { notEmpty }
   { gr }
   { lt }
   { app }
   { partGt, partGt[Ite][True][Ite] }
   { partLt[Ite][True][Ite], partLt }
   { part, quicksort }
   { goal }

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {notEmpty}, {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {notEmpty}, {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: notEmpty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {notEmpty}, {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: ?, size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: gr
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: ?, size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: gr
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](gr(x3, z), z, 1 + x3 + xs'))) :|: z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x1, z - 1), 1 + (z - 1), 1 + (1 + x1) + xs) :|: z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {lt}, {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: ?, size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](lt(x3, z), z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs) :|: xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: ?, size: O(n^1) [z + z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {partGt,partGt[Ite][True][Ite]}, {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: ?, size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: ?, size: O(n^1) [z'']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 3 }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(partGt[Ite][True][Ite](s', z, 1 + x3 + xs'))) :|: s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + x1) + xs) :|: s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](2, 0, 1 + (1 + x2) + xs) :|: xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 4 + x3 + xs' }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 4 + x3 + xs' }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {partLt[Ite][True][Ite],partLt}, {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']
  partLt[Ite][True][Ite]: runtime: ?, size: O(n^1) [z'']
  partLt: runtime: ?, size: O(n^1) [z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: partLt[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + z''

Computed RUNTIME bound using CoFloCo for: partLt
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + z'

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 4 + x3 + xs' }-> app(quicksort(partLt[Ite][True][Ite](s2, z, 1 + x3 + xs')), 1 + z + quicksort(s9)) :|: s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs) :|: s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](1, 0, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(z', xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']
  partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z'']
  partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z']

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 10 + 2*x3 + 2*xs' }-> app(quicksort(s16), 1 + z + quicksort(s9)) :|: s16 >= 0, s16 <= 1 + x3 + xs', s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 8 + x'' + xs }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + x'') + xs, s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 6 + z' }-> s13 :|: s13 >= 0, s13 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 7 + x + xs }-> s14 :|: s14 >= 0, s14 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 7 + x + xs }-> s15 :|: s15 >= 0, s15 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> s18 :|: s18 >= 0, s18 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> 1 + x + s17 :|: s17 >= 0, s17 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']
  partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z'']
  partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z']

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 10 + 2*x3 + 2*xs' }-> app(quicksort(s16), 1 + z + quicksort(s9)) :|: s16 >= 0, s16 <= 1 + x3 + xs', s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 8 + x'' + xs }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + x'') + xs, s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 6 + z' }-> s13 :|: s13 >= 0, s13 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 7 + x + xs }-> s14 :|: s14 >= 0, s14 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 7 + x + xs }-> s15 :|: s15 >= 0, s15 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> s18 :|: s18 >= 0, s18 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> 1 + x + s17 :|: s17 >= 0, s17 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']
  partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z'']
  partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z']
  part: runtime: ?, size: EXP
  quicksort: runtime: ?, size: INF

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(z) :|: z >= 0
   gr(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: z' >= 0, z = 0
   gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   lt(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   part(z, z') -{ 10 + 2*x3 + 2*xs' }-> app(quicksort(s16), 1 + z + quicksort(s9)) :|: s16 >= 0, s16 <= 1 + x3 + xs', s9 >= 0, s9 <= 1 + x3 + xs', s2 >= 0, s2 <= 2, s' >= 0, s' <= 2, z' = 1 + x3 + xs', z >= 0, xs' >= 0, x3 >= 0
   part(z, z') -{ 3 }-> app(quicksort(0), 1 + z + quicksort(0)) :|: z >= 0, z' = 0
   partGt(z, z') -{ 3 + x1 + xs }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + x1) + xs, s >= 0, s <= 2, z - 1 >= 0, xs >= 0, x1 >= 0, z' = 1 + (1 + x1) + xs
   partGt(z, z') -{ 1 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z >= 0
   partGt(z, z') -{ 3 + x2 + xs }-> s7 :|: s7 >= 0, s7 <= 1 + (1 + x2) + xs, xs >= 0, z' = 1 + (1 + x2) + xs, z = 0, x2 >= 0
   partGt(z, z') -{ 2 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partGt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> s11 :|: s11 >= 0, s11 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 1 + xs }-> 1 + x + s10 :|: s10 >= 0, s10 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 8 + x'' + xs }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + x'') + xs, s1 >= 0, s1 <= 2, xs >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   partLt(z, z') -{ 6 + z' }-> s13 :|: s13 >= 0, s13 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0
   partLt(z, z') -{ 7 + x + xs }-> s14 :|: s14 >= 0, s14 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, x >= 0, z = 0
   partLt(z, z') -{ 7 + x + xs }-> s15 :|: s15 >= 0, s15 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   partLt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> s18 :|: s18 >= 0, s18 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 6 + xs }-> 1 + x + s17 :|: s17 >= 0, s17 <= xs, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part,quicksort}, {goal}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  gr: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  app: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  partGt: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  partGt[Ite][True][Ite]: runtime: O(n^1) [z''], size: O(n^1) [z'']
  partLt[Ite][True][Ite]: runtime: O(n^1) [5 + z''], size: O(n^1) [z'']
  partLt: runtime: O(n^1) [6 + z'], size: O(n^1) [z']
  part: runtime: INF, size: EXP
  quicksort: runtime: ?, size: INF

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

(65) 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:

   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   partLt[Ite][True][Ite](v0, v1, v2) -> null_partLt[Ite][True][Ite] [0]
   partGt[Ite][True][Ite](v0, v1, v2) -> null_partGt[Ite][True][Ite] [0]
   quicksort(v0) -> null_quicksort [0]
   partLt(v0, v1) -> null_partLt [0]
   partGt(v0, v1) -> null_partGt [0]
   app(v0, v1) -> null_app [0]
   notEmpty(v0) -> null_notEmpty [0]

And the following fresh constants:    null_lt, null_gr, null_partLt[Ite][True][Ite], null_partGt[Ite][True][Ite], null_quicksort, null_partLt, null_partGt, null_app, null_notEmpty

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

(66)
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:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x', xs)) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   partLt(x', Cons(x, xs)) -> partLt[Ite][True][Ite](lt(x, x'), x', Cons(x, xs)) [1]
   partLt(x, Nil) -> Nil [1]
   partGt(x', Cons(x, xs)) -> partGt[Ite][True][Ite](gr(x, x'), x', Cons(x, xs)) [1]
   partGt(x, Nil) -> Nil [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   app(Nil, ys) -> ys [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   part(x, xs) -> app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs)))) [1]
   goal(xs) -> quicksort(xs) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   gr(S(x), S(y)) -> gr(x, y) [0]
   gr(0, y) -> False [0]
   gr(S(x), 0) -> True [0]
   partLt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partLt(x', xs)) [0]
   partGt[Ite][True][Ite](True, x', Cons(x, xs)) -> Cons(x, partGt(x', xs)) [0]
   partLt[Ite][True][Ite](False, x', Cons(x, xs)) -> partLt(x', xs) [0]
   partGt[Ite][True][Ite](False, x', Cons(x, xs)) -> partGt(x', xs) [0]
   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   partLt[Ite][True][Ite](v0, v1, v2) -> null_partLt[Ite][True][Ite] [0]
   partGt[Ite][True][Ite](v0, v1, v2) -> null_partGt[Ite][True][Ite] [0]
   quicksort(v0) -> null_quicksort [0]
   partLt(v0, v1) -> null_partLt [0]
   partGt(v0, v1) -> null_partGt [0]
   app(v0, v1) -> null_app [0]
   notEmpty(v0) -> null_notEmpty [0]

The TRS has the following type information:
quicksort :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
Cons :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
part :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
Nil :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
partLt :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
partLt[Ite][True][Ite] :: True:False:null_lt:null_gr:null_notEmpty -> S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
lt :: S:0 -> S:0 -> True:False:null_lt:null_gr:null_notEmpty
partGt :: S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
partGt[Ite][True][Ite] :: True:False:null_lt:null_gr:null_notEmpty -> S:0 -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
gr :: S:0 -> S:0 -> True:False:null_lt:null_gr:null_notEmpty
app :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
notEmpty :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> True:False:null_lt:null_gr:null_notEmpty
True :: True:False:null_lt:null_gr:null_notEmpty
False :: True:False:null_lt:null_gr:null_notEmpty
goal :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app -> Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
S :: S:0 -> S:0
0 :: S:0
null_lt :: True:False:null_lt:null_gr:null_notEmpty
null_gr :: True:False:null_lt:null_gr:null_notEmpty
null_partLt[Ite][True][Ite] :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
null_partGt[Ite][True][Ite] :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
null_quicksort :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
null_partLt :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
null_partGt :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
null_app :: Cons:Nil:null_partLt[Ite][True][Ite]:null_partGt[Ite][True][Ite]:null_quicksort:null_partLt:null_partGt:null_app
null_notEmpty :: True:False:null_lt:null_gr:null_notEmpty

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

(67) 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
   null_lt => 0
   null_gr => 0
   null_partLt[Ite][True][Ite] => 0
   null_partGt[Ite][True][Ite] => 0
   null_quicksort => 0
   null_partLt => 0
   null_partGt => 0
   null_app => 0
   null_notEmpty => 0

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

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

   app(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + x + app(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   goal(z) -{ 1 }-> quicksort(xs) :|: xs >= 0, z = xs
   gr(z, z') -{ 0 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 0 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 0 }-> 1 :|: y >= 0, z = 0, z' = y
   gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   notEmpty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   part(z, z') -{ 1 }-> app(quicksort(partLt(x, xs)), 1 + x + quicksort(partGt(x, xs))) :|: xs >= 0, x >= 0, z' = xs, z = x
   partGt(z, z') -{ 1 }-> partGt[Ite][True][Ite](gr(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   partGt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   partGt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> partGt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   partGt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partGt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
   partLt(z, z') -{ 1 }-> partLt[Ite][True][Ite](lt(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   partLt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   partLt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> partLt(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   partLt[Ite][True][Ite](z, z', z'') -{ 0 }-> 1 + x + partLt(x', xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x' + xs) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   quicksort(z) -{ 1 }-> 1 + x + 0 :|: x >= 0, z = 1 + x + 0

Only complete derivations are relevant for the runtime complexity.

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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   goal(z0) -> quicksort(z0)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   <'(0, S(z0)) -> c1
   <'(z0, 0) -> c2
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   >'(0, z0) -> c4
   >'(S(z0), 0) -> c5
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   QUICKSORT(Cons(z0, Nil)) -> c11
   QUICKSORT(Nil) -> c12
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   APP(Nil, z0) -> c18
   NOTEMPTY(Cons(z0, z1)) -> c19
   NOTEMPTY(Nil) -> c20
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   GOAL(z0) -> c23(QUICKSORT(z0))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   QUICKSORT(Cons(z0, Nil)) -> c11
   QUICKSORT(Nil) -> c12
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   APP(Nil, z0) -> c18
   NOTEMPTY(Cons(z0, z1)) -> c19
   NOTEMPTY(Nil) -> c20
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   GOAL(z0) -> c23(QUICKSORT(z0))
K tuples:none
Defined Rule Symbols:   quicksort_1, partLt_2, partGt_2, app_2, notEmpty_1, part_2, goal_1, <_2, >_2, partLt[Ite][True][Ite]_3, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, NOTEMPTY_1, PART_2, GOAL_1

Compound Symbols:   c_1, c1, c2, c3_1, c4, c5, c6_1, c7_1, c8_1, c9_1, c10_1, c11, c12, c13_2, c14, c15_2, c16, c17_1, c18, c19, c20, c21_3, c22_3, c23_1


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

(71) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c23(QUICKSORT(z0))
Removed 9 trailing nodes:
   >'(0, z0) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c19
   QUICKSORT(Cons(z0, Nil)) -> c11
   >'(S(z0), 0) -> c5
   APP(Nil, z0) -> c18
   <'(z0, 0) -> c2
   NOTEMPTY(Nil) -> c20
   QUICKSORT(Nil) -> c12
   <'(0, S(z0)) -> c1

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   goal(z0) -> quicksort(z0)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
K tuples:none
Defined Rule Symbols:   quicksort_1, partLt_2, partGt_2, app_2, notEmpty_1, part_2, goal_1, <_2, >_2, partLt[Ite][True][Ite]_3, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3


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

(73) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> quicksort(z0)

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
K tuples:none
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3


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

(75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
We considered the (Usable) Rules:
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1] + x_1 + x_2
   POL(<'(x_1, x_2)) = 0
   POL(>(x_1, x_2)) = [1] + x_1 + x_2
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2)) = x_2
   POL(PARTGT(x_1, x_2)) = 0
   POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0
   POL(PARTLT(x_1, x_2)) = 0
   POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = [1] + x_1
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16) = 0
   POL(c17(x_1)) = x_1
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(part(x_1, x_2)) = [1] + x_1 + x_2
   POL(partGt(x_1, x_2)) = x_2
   POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(partLt(x_1, x_2)) = x_2
   POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(quicksort(x_1)) = 0

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
S tuples:
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3


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

(77) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
S tuples:
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PARTGT(z0, Nil) -> c16
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1] + x_1 + x_2
   POL(<'(x_1, x_2)) = 0
   POL(>(x_1, x_2)) = [1]
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2)) = [1] + x_2
   POL(PARTGT(x_1, x_2)) = [1]
   POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = x_1
   POL(PARTLT(x_1, x_2)) = 0
   POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = [1] + x_1
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16) = 0
   POL(c17(x_1)) = x_1
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(part(x_1, x_2)) = [1] + x_1 + x_2
   POL(partGt(x_1, x_2)) = x_2
   POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(partLt(x_1, x_2)) = x_2
   POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(quicksort(x_1)) = 0

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
S tuples:
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTGT(z0, Nil) -> c16
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PARTLT(z0, Nil) -> c14
We considered the (Usable) Rules:
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1] + x_1 + x_2
   POL(<'(x_1, x_2)) = 0
   POL(>(x_1, x_2)) = [1] + x_1 + x_2
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2)) = [1] + x_2
   POL(PARTGT(x_1, x_2)) = 0
   POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0
   POL(PARTLT(x_1, x_2)) = [1]
   POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = [1]
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = [1] + x_1
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14) = 0
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c16) = 0
   POL(c17(x_1)) = x_1
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(part(x_1, x_2)) = [1] + x_1 + x_2
   POL(partGt(x_1, x_2)) = x_2
   POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(partLt(x_1, x_2)) = x_2
   POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(quicksort(x_1)) = 0

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
S tuples:
   PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0))
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTGT(z0, Nil) -> c16
   PARTLT(z0, Nil) -> c14
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c13_2, c14, c15_2, c16, c17_1, c21_3, c22_3


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) by 
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(z0, Nil) -> c14
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   PARTGT(z0, Nil) -> c16
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0))
S tuples:
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTGT(z0, Nil) -> c16
   PARTLT(z0, Nil) -> c14
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTLT_2, PARTGT_2, APP_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c14, c15_2, c16, c17_1, c21_3, c22_3, c13_2


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

(85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   PARTGT(z0, Nil) -> c16
   PARTLT(z0, Nil) -> c14

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0))
S tuples:
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)), <'(0, S(z0)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)), <'(z0, 0))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTGT_2, APP_2, PART_2, PARTLT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c15_2, c17_1, c21_3, c22_3, c13_2


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

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
S tuples:
   PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, PARTGT_2, APP_2, PART_2, PARTLT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c15_2, c17_1, c21_3, c22_3, c13_2, c13_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) by 
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)), >'(0, z0))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)), >'(S(z0), 0))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)), >'(0, z0))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)), >'(S(z0), 0))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)), >'(0, z0))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)), >'(S(z0), 0))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c21_3, c22_3, c13_2, c13_1, c15_2


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c21_3, c22_3, c13_2, c13_1, c15_2, c15_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) by 
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c21(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c21(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c21(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c21(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partLt(z0, Nil)), PARTLT(z0, Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c22_3, c13_2, c13_1, c15_2, c15_1, c21_3


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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c21(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c21(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partLt(z0, Nil)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c22_3, c13_2, c13_1, c15_2, c15_1, c21_3, c21_2


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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PART_2, PARTLT_2, PARTGT_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c22_3, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) by 
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c22(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c22(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c22(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c22(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partGt(z0, Nil)), PARTGT(z0, Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3


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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c22(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))), QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c22(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))), QUICKSORT(partGt(z0, Nil)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c22_2


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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_2, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c2_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PARTLT(S(z1), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2)), <'(S(z0), S(z1))) by 
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2


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

(107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
We considered the (Usable) Rules:
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = 0
   POL(<'(x_1, x_2)) = 0
   POL(>(x_1, x_2)) = x_1
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2)) = [1] + x_2
   POL(PARTGT(x_1, x_2)) = 0
   POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0
   POL(PARTLT(x_1, x_2)) = [1]
   POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = [1]
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = 0
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(part(x_1, x_2)) = [1] + x_1 + x_2
   POL(partGt(x_1, x_2)) = x_2
   POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(partLt(x_1, x_2)) = x_2
   POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(quicksort(x_1)) = 0

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_2, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PARTGT(S(z1), Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(z1), Cons(S(z0), x2)), >'(S(z0), S(z1))) by 
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2, c15_2


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

(111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
We considered the (Usable) Rules:
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
   partLt(z0, Nil) -> Nil
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = 0
   POL(<'(x_1, x_2)) = 0
   POL(>(x_1, x_2)) = 0
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2)) = [1] + x_2
   POL(PARTGT(x_1, x_2)) = [1]
   POL(PARTGT[ITE][TRUE][ITE](x_1, x_2, x_3)) = [1]
   POL(PARTLT(x_1, x_2)) = 0
   POL(PARTLT[ITE][TRUE][ITE](x_1, x_2, x_3)) = 0
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = 0
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c15(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(part(x_1, x_2)) = [1] + x_1 + x_2
   POL(partGt(x_1, x_2)) = x_2
   POL(partGt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(partLt(x_1, x_2)) = x_2
   POL(partLt[Ite][True][Ite](x_1, x_2, x_3)) = x_3
   POL(quicksort(x_1)) = 0

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2, c15_2


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by 
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c21_3, c1_1, c22_3, c2_1, c13_2, c15_2


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by 
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c1(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) by 
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) by 
   PART(z0, Nil) -> c1(QUICKSORT(Nil))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1))
   PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3


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

(121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(z0, Nil) -> c1(QUICKSORT(Nil))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) by 
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(x0, Nil) -> c1(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil)))))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(x0, Nil) -> c1(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil)))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3


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

(125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil) -> c1(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil)))))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c1_1, c22_3, c2_1, c13_2, c15_2, c21_3


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

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c1(QUICKSORT(partLt(z0, Nil))) by 
   PART(z0, Nil) -> c1(QUICKSORT(Nil))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c1(QUICKSORT(Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c22_3, c2_1, c13_2, c15_2, c21_3, c1_1


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

(129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(z0, Nil) -> c1(QUICKSORT(Nil))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c22_3, c2_1, c13_2, c15_2, c21_3, c1_1


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

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt(z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by 
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c22_3, c2_1, c13_2, c15_2, c21_3, c1_1


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(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt(z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by 
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3


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

(135) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c2(APP(quicksort(partLt(z0, Nil)), Cons(z0, quicksort(Nil)))) by 
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) by 
   PART(z0, Nil) -> c2(QUICKSORT(Nil))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3


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

(139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(z0, Nil) -> c2(QUICKSORT(Nil))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil)))))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(partGt(z0, Nil))))) by 
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(x0, Nil) -> c2(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil)))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(x0, Nil) -> c2(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil)))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3


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

(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil) -> c2(APP(Nil, Cons(x0, quicksort(partGt(x0, Nil)))))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c2_1, c13_2, c15_2, c21_3, c1_1, c22_3


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil) -> c2(QUICKSORT(partGt(z0, Nil))) by 
   PART(z0, Nil) -> c2(QUICKSORT(Nil))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(z0, Nil) -> c2(QUICKSORT(Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1


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

(147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(z0, Nil) -> c2(QUICKSORT(Nil))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1


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

(149) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) by 
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2

Compound Symbols:   c_1, c3_1, c7_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1


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

(151) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) by 
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Cons(0, x2)) -> c21(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partLt(z0, Cons(0, x2))), PARTLT(z0, Cons(0, x2))) by PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(0, Cons(S(z0), x2)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partLt(0, Cons(S(z0), x2))), PARTLT(0, Cons(S(z0), x2))) by PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c8_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1


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

(159) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) by 
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, PARTGT[ITE][TRUE][ITE]_3, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c9_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1


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

(163) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) by 
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3

Compound Symbols:   c_1, c3_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace <'(S(z0), S(z1)) -> c(<'(z0, z1)) by 
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)), <'(S(0), S(S(z0))))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)), <'(S(z0), S(0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2

Compound Symbols:   c3_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1


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

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2

Compound Symbols:   c3_1, c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace >'(S(z0), S(z1)) -> c3(>'(z0, z1)) by 
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)), >'(S(0), S(z0)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(0)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2

Compound Symbols:   c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1


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

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2

Compound Symbols:   c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1


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

(173) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt(z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
S tuples:
   APP(Cons(z0, z1), z2) -> c17(APP(z1, z2))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, APP_2, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2

Compound Symbols:   c10_1, c17_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1


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

(175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) by 
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1, c17_1


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

(177) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partLt(S(z1), Cons(S(z0), x2))), PARTLT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1, c17_1


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(S(z0), Cons(0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partLt(S(z0), Cons(0, x2))), PARTLT(S(z0), Cons(0, x2))) by PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c21_3, c1_1, c22_3, c2_1, c6_1, c7_1, c8_1, c9_1, c_1, c3_1, c17_1


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

(181) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(0, Cons(z0, x2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partLt(0, Cons(z0, x2))), PARTLT(0, Cons(z0, x2))) by PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c1_1, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1


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

(183) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil) -> c1(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c1_1, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c1_1, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1


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

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) by PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt[Ite][True][Ite](>(z0, z1), S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(191) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Cons(0, x2)) -> c22(APP(quicksort(partLt(z0, Cons(0, x2))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, x2))))), QUICKSORT(partGt(z0, Cons(0, x2))), PARTGT(z0, Cons(0, x2))) by PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(0, Cons(S(z0), x2)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), x2))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), x2))))), QUICKSORT(partGt(0, Cons(S(z0), x2))), PARTGT(0, Cons(S(z0), x2))) by PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(197) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt(z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2))) by PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(199) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(S(z1), Cons(S(z0), x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z0, z1), S(z1), Cons(S(z0), x2))), Cons(S(z1), quicksort(partGt(S(z1), Cons(S(z0), x2))))), QUICKSORT(partGt(S(z1), Cons(S(z0), x2))), PARTGT(S(z1), Cons(S(z0), x2))) by PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(201) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(S(z0), Cons(0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, x2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, x2))))), QUICKSORT(partGt(S(z0), Cons(0, x2))), PARTGT(S(z0), Cons(0, x2))) by PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1)))
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c22_3, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1


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

(203) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(0, Cons(z0, x2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, x2))), Cons(0, quicksort(partGt(0, Cons(z0, x2))))), QUICKSORT(partGt(0, Cons(z0, x2))), PARTGT(0, Cons(z0, x2))) by PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1)))
----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1, c22_3


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

(205) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil) -> c2(APP(quicksort(partLt(x0, Nil)), Cons(x0, Nil))) by PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil)))
----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1, c22_3


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

(207) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil)))) by PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil)))
----------------------------------------

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2))
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))
   partLt(z0, Nil) -> Nil
   partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2))
   partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2)
   part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1))))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))
   partGt(z0, Nil) -> Nil
   partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2))
   partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2)
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, quicksort(Nil))))
   PARTLT[ITE][TRUE][ITE](True, S(x0), Cons(0, x1)) -> c6(PARTLT(S(x0), x1))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c6(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1)) -> c6(PARTLT(S(S(x0)), x1))
   PARTLT[ITE][TRUE][ITE](False, 0, Cons(x0, x1)) -> c7(PARTLT(0, x1))
   PARTLT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(PARTLT(S(S(x0)), x2))
   PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1)) -> c7(PARTLT(S(0), x1))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c21(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, z0), z0, Cons(0, z1))), PARTLT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c21(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z0), 0), 0, Cons(S(z0), z1))), PARTLT(0, Cons(S(z0), z1)))
   PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(x0), x1)) -> c8(PARTGT(0, x1))
   PARTGT[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2)) -> c8(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(x0)), x1)) -> c8(PARTGT(S(0), x1))
   PART(z0, Cons(z1, z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), PARTLT(z0, Cons(z1, z2)))
   PARTGT[ITE][TRUE][ITE](False, x0, Cons(0, x1)) -> c9(PARTGT(x0, x1))
   PARTGT[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c9(PARTGT(S(S(x0)), x2))
   PARTGT[ITE][TRUE][ITE](False, S(x0), Cons(S(0), x1)) -> c9(PARTGT(S(x0), x1))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
   PART(S(z0), Cons(S(z1), z2)) -> c21(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partLt[Ite][True][Ite](<(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTLT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(0, S(z0)), S(z0), Cons(0, z1))), PARTLT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c21(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partLt[Ite][True][Ite](<(z0, 0), 0, Cons(z0, z1))), PARTLT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c1(APP(quicksort(Nil), Cons(z0, Nil)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt(S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt[Ite][True][Ite](>(z1, z0), S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(z0, Cons(0, z1)) -> c22(APP(quicksort(partLt(z0, Cons(0, z1))), Cons(z0, quicksort(partGt[Ite][True][Ite](False, z0, Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, z0), z0, Cons(0, z1))), PARTGT(z0, Cons(0, z1)))
   PART(0, Cons(S(z0), z1)) -> c22(APP(quicksort(partLt(0, Cons(S(z0), z1))), Cons(0, quicksort(partGt[Ite][True][Ite](True, 0, Cons(S(z0), z1))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z0), 0), 0, Cons(S(z0), z1))), PARTGT(0, Cons(S(z0), z1)))
   PART(z0, Cons(z1, z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2))), Cons(z0, quicksort(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))))), QUICKSORT(partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2))), PARTGT(z0, Cons(z1, z2)))
   PART(S(z0), Cons(S(z1), z2)) -> c22(APP(quicksort(partLt[Ite][True][Ite](<(z1, z0), S(z0), Cons(S(z1), z2))), Cons(S(z0), quicksort(partGt(S(z0), Cons(S(z1), z2))))), QUICKSORT(partGt[Ite][True][Ite](>(S(z1), S(z0)), S(z0), Cons(S(z1), z2))), PARTGT(S(z0), Cons(S(z1), z2)))
   PART(S(z0), Cons(0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](True, S(z0), Cons(0, z1))), Cons(S(z0), quicksort(partGt(S(z0), Cons(0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(0, S(z0)), S(z0), Cons(0, z1))), PARTGT(S(z0), Cons(0, z1)))
   PART(0, Cons(z0, z1)) -> c22(APP(quicksort(partLt[Ite][True][Ite](False, 0, Cons(z0, z1))), Cons(0, quicksort(partGt(0, Cons(z0, z1))))), QUICKSORT(partGt[Ite][True][Ite](>(z0, 0), 0, Cons(z0, z1))), PARTGT(0, Cons(z0, z1)))
   PART(z0, Nil) -> c2(APP(quicksort(Nil), Cons(z0, Nil)))
S tuples:
   PARTLT(S(z0), Cons(0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(z0), Cons(0, x2)))
   PARTLT(0, Cons(z0, x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, 0, Cons(z0, x2)))
   PARTGT(z0, Cons(0, x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, z0, Cons(0, x2)))
   PARTGT(0, Cons(S(z0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, 0, Cons(S(z0), x2)))
   PARTLT(S(S(z1)), Cons(S(S(z0)), x2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), <'(S(S(z0)), S(S(z1))))
   PARTGT(S(S(z1)), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z0, z1), S(S(z1)), Cons(S(S(z0)), x2)), >'(S(S(z0)), S(S(z1))))
   PARTLT(S(S(z0)), Cons(S(0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2)))
   PARTLT(S(0), Cons(S(z0), x2)) -> c13(PARTLT[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2)))
   PARTGT(S(z0), Cons(S(0), x2)) -> c15(PARTGT[ITE][TRUE][ITE](False, S(z0), Cons(S(0), x2)))
   PARTGT(S(0), Cons(S(S(z0)), x2)) -> c15(PARTGT[ITE][TRUE][ITE](True, S(0), Cons(S(S(z0)), x2)))
   APP(Cons(z0, Cons(y0, y1)), z2) -> c17(APP(Cons(y0, y1), z2))
K tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2)))
   PARTLT(S(x0), Cons(S(x1), x2)) -> c13(<'(S(x1), S(x0)))
   PARTGT(S(x0), Cons(S(x1), x2)) -> c15(>'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, >_2, quicksort_1, partLt_2, partLt[Ite][True][Ite]_3, part_2, app_2, partGt_2, partGt[Ite][True][Ite]_3

Defined Pair Symbols:   QUICKSORT_1, PARTLT_2, PARTGT_2, PART_2, PARTLT[ITE][TRUE][ITE]_3, PARTGT[ITE][TRUE][ITE]_3, <'_2, >'_2, APP_2

Compound Symbols:   c10_1, c13_1, c15_1, c13_2, c15_2, c2_1, c6_1, c7_1, c21_3, c8_1, c9_1, c_1, c3_1, c17_1, c1_1, c22_3