KILLED
proof of input_WUxBLZlidl.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 439 ms]
(2) CpxRelTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxTypedWeightedCompleteTrs
        (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxRNTS
        (25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 142 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 358 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 406 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 403 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 23.7 s]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 5154 ms]
        (56) CpxRNTS
(57) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(58) CdtProblem
    (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (60) CdtProblem
    (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 47 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms]
    (76) CdtProblem
    (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 74 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 72 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 118 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (120) CdtProblem
    (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms]
    (130) CdtProblem
    (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 88 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (134) CdtProblem
    (135) CdtLeafRemovalProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtLeafRemovalProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 121 ms]
    (166) CdtProblem
    (167) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 123 ms]
    (168) CdtProblem
    (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtRewritingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (184) CdtProblem
    (185) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtInstantiationProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtRhsSimplificationProcessorProof [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
    (209) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) CdtProblem
    (211) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (212) CdtProblem
    (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (214) CdtProblem
    (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (216) CdtProblem
    (217) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 182 ms]
    (218) CdtProblem
    (219) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (220) CdtProblem
    (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (222) CdtProblem
    (223) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (224) CdtProblem
    (225) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) 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, Cons(x', xs)), Cons(x, Nil), Nil)
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   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
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2)
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2)

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, Cons(x', xs)), Cons(x, Nil), Nil)
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   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
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2)
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2)

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

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

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


The TRS R consists of the following rules:

   quicksort(Cons(x, Cons(x', xs))) -> part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   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
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2)
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2)

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

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

(6)
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, Cons(x', xs)), Cons(x, Nil), Nil)
   quicksort(Cons(x, Nil)) -> Cons(x, Nil)
   quicksort(Nil) -> Nil
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   app(Nil, ys) -> ys
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   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
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2)
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2)

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, Cons(x', xs)), Cons(x, Nil), Nil) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2) [1]
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2)) [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]
   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]
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2) [0]
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2) [0]

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

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

> => gr
< => lt

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

(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, Cons(x', xs)), Cons(x, Nil), Nil) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](gr(x', x), x', Cons(x, xs), xs1, xs2) [1]
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2)) [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]
   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]
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2) [0]
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](lt(x', x), x', Cons(x, xs), xs1, xs2) [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, Cons(x', xs)), Cons(x, Nil), Nil) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](gr(x', x), x', Cons(x, xs), xs1, xs2) [1]
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2)) [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]
   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]
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2) [0]
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](lt(x', x), x', Cons(x, xs), xs1, xs2) [0]

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

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

(13) 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]
   part[Ite][True][Ite](v0, v1, v2, v3, v4) -> null_part[Ite][True][Ite] [0]
   quicksort(v0) -> null_quicksort [0]
   part(v0, v1, v2, v3) -> null_part [0]
   app(v0, v1) -> null_app [0]
   notEmpty(v0) -> null_notEmpty [0]

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

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

(14)
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, Cons(x', xs)), Cons(x, Nil), Nil) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](gr(x', x), x', Cons(x, xs), xs1, xs2) [1]
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2)) [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]
   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]
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2) [0]
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](lt(x', x), x', Cons(x, xs), xs1, xs2) [0]
   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   part[Ite][True][Ite](v0, v1, v2, v3, v4) -> null_part[Ite][True][Ite] [0]
   quicksort(v0) -> null_quicksort [0]
   part(v0, v1, v2, v3) -> null_part [0]
   app(v0, v1) -> null_app [0]
   notEmpty(v0) -> null_notEmpty [0]

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

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

(15) 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_part[Ite][True][Ite] => 0
   null_quicksort => 0
   null_part => 0
   null_app => 0
   null_notEmpty => 0

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

(16)
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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(x', x), x', 1 + x + xs, xs1, xs2) :|: xs >= 0, xs2 >= 0, z' = 1 + x + xs, z1 = xs2, x' >= 0, x >= 0, z'' = xs1, xs1 >= 0, z = x'
   part(z, z', z'', z1) -{ 1 }-> app(quicksort(xs1), quicksort(xs2)) :|: xs2 >= 0, z1 = xs2, x >= 0, z'' = xs1, xs1 >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(x', xs, 1 + x + xs1, xs2) :|: z = 2, z' = x', xs >= 0, xs2 >= 0, z2 = xs2, x' >= 0, x >= 0, z1 = xs1, xs1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(x', x) + x' + (1 + x + xs) + xs1 + xs2 :|: z' = x', xs >= 0, xs2 >= 0, z = 1, z2 = xs2, x' >= 0, x >= 0, z1 = xs1, xs1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: 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.

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

(17) 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
   part_4
   app_2
   lt_2
   gr_2
   part[Ite][True][Ite]_5

Due to the following rules being added:

   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   part[Ite][True][Ite](v0, v1, v2, v3, v4) -> Nil [0]
   quicksort(v0) -> Nil [0]
   part(v0, v1, v2, v3) -> Nil [0]
   app(v0, v1) -> Nil [0]

And the following fresh constants:    null_lt, null_gr

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

(18)
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, Cons(x', xs)), Cons(x, Nil), Nil) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](gr(x', x), x', Cons(x, xs), xs1, xs2) [1]
   part(x, Nil, xs1, xs2) -> app(quicksort(xs1), quicksort(xs2)) [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]
   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]
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2) [0]
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](lt(x', x), x', Cons(x, xs), xs1, xs2) [0]
   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   part[Ite][True][Ite](v0, v1, v2, v3, v4) -> Nil [0]
   quicksort(v0) -> Nil [0]
   part(v0, v1, v2, v3) -> Nil [0]
   app(v0, v1) -> Nil [0]

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

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

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

(20)
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, Cons(x', xs)), Cons(x, Nil), Nil) [1]
   quicksort(Cons(x, Nil)) -> Cons(x, Nil) [1]
   quicksort(Nil) -> Nil [1]
   part(S(x''), Cons(S(y'), xs), xs1, xs2) -> part[Ite][True][Ite](gr(x'', y'), S(x''), Cons(S(y'), xs), xs1, xs2) [1]
   part(0, Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](False, 0, Cons(x, xs), xs1, xs2) [1]
   part(S(x1), Cons(0, xs), xs1, xs2) -> part[Ite][True][Ite](True, S(x1), Cons(0, xs), xs1, xs2) [1]
   part(x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite](null_gr, x', Cons(x, xs), xs1, xs2) [1]
   part(x, Nil, Cons(x2, Cons(x''', xs')), Cons(x4, Cons(x'1, xs''))) -> app(part(x2, Cons(x2, Cons(x''', xs')), Cons(x2, Nil), Nil), part(x4, Cons(x4, Cons(x'1, xs'')), Cons(x4, Nil), Nil)) [3]
   part(x, Nil, Cons(x2, Cons(x''', xs')), Cons(x5, Nil)) -> app(part(x2, Cons(x2, Cons(x''', xs')), Cons(x2, Nil), Nil), Cons(x5, Nil)) [3]
   part(x, Nil, Cons(x2, Cons(x''', xs')), Nil) -> app(part(x2, Cons(x2, Cons(x''', xs')), Cons(x2, Nil), Nil), Nil) [3]
   part(x, Nil, Cons(x2, Cons(x''', xs')), xs2) -> app(part(x2, Cons(x2, Cons(x''', xs')), Cons(x2, Nil), Nil), Nil) [2]
   part(x, Nil, Cons(x3, Nil), Cons(x6, Cons(x'2, xs3))) -> app(Cons(x3, Nil), part(x6, Cons(x6, Cons(x'2, xs3)), Cons(x6, Nil), Nil)) [3]
   part(x, Nil, Cons(x3, Nil), Cons(x7, Nil)) -> app(Cons(x3, Nil), Cons(x7, Nil)) [3]
   part(x, Nil, Cons(x3, Nil), Nil) -> app(Cons(x3, Nil), Nil) [3]
   part(x, Nil, Cons(x3, Nil), xs2) -> app(Cons(x3, Nil), Nil) [2]
   part(x, Nil, Nil, Cons(x8, Cons(x'3, xs4))) -> app(Nil, part(x8, Cons(x8, Cons(x'3, xs4)), Cons(x8, Nil), Nil)) [3]
   part(x, Nil, Nil, Cons(x9, Nil)) -> app(Nil, Cons(x9, Nil)) [3]
   part(x, Nil, Nil, Nil) -> app(Nil, Nil) [3]
   part(x, Nil, Nil, xs2) -> app(Nil, Nil) [2]
   part(x, Nil, xs1, Cons(x10, Cons(x'4, xs5))) -> app(Nil, part(x10, Cons(x10, Cons(x'4, xs5)), Cons(x10, Nil), Nil)) [2]
   part(x, Nil, xs1, Cons(x11, Nil)) -> app(Nil, Cons(x11, Nil)) [2]
   part(x, Nil, xs1, Nil) -> app(Nil, Nil) [2]
   part(x, Nil, xs1, xs2) -> app(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]
   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]
   part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) -> part(x', xs, Cons(x, xs1), xs2) [0]
   part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) -> part[Ite][True][Ite][False][Ite](lt(x', x), x', Cons(x, xs), xs1, xs2) [0]
   lt(v0, v1) -> null_lt [0]
   gr(v0, v1) -> null_gr [0]
   part[Ite][True][Ite](v0, v1, v2, v3, v4) -> Nil [0]
   quicksort(v0) -> Nil [0]
   part(v0, v1, v2, v3) -> Nil [0]
   app(v0, v1) -> Nil [0]

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

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

(21) 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

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

(22)
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
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(x'', y'), 1 + x'', 1 + (1 + y') + xs, xs1, xs2) :|: z = 1 + x'', xs >= 0, xs2 >= 0, z' = 1 + (1 + y') + xs, z1 = xs2, y' >= 0, z'' = xs1, xs1 >= 0, x'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + x1, 1 + 0 + xs, xs1, xs2) :|: xs >= 0, x1 >= 0, xs2 >= 0, z' = 1 + 0 + xs, z1 = xs2, z = 1 + x1, z'' = xs1, xs1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, xs1, xs2) :|: xs >= 0, xs2 >= 0, z' = 1 + x + xs, z1 = xs2, x >= 0, z'' = xs1, xs1 >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, x', 1 + x + xs, xs1, xs2) :|: xs >= 0, xs2 >= 0, z' = 1 + x + xs, z1 = xs2, x' >= 0, x >= 0, z'' = xs1, xs1 >= 0, z = x'
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, x >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, z = x, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), x >= 0, xs' >= 0, x''' >= 0, z = x, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: xs2 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z1 = xs2, x >= 0, xs' >= 0, x''' >= 0, z = x, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + x5 + 0) :|: x5 >= 0, z1 = 1 + x5 + 0, z'' = 1 + x2 + (1 + x''' + xs'), x >= 0, xs' >= 0, x''' >= 0, z = x, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), x >= 0, z'' = xs1, xs1 >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, x >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, x >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, xs2 >= 0, z1 = xs2, x >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, x >= 0, z'' = xs1, xs1 >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: xs2 >= 0, z1 = xs2, x >= 0, z'' = xs1, xs1 >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + x11 + 0) :|: z1 = 1 + x11 + 0, x >= 0, z'' = xs1, x11 >= 0, xs1 >= 0, z = x, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + x9 + 0) :|: z'' = 0, x >= 0, z1 = 1 + x9 + 0, z = x, x9 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + x3 + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: x >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z = x, z'' = 1 + x3 + 0, x3 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + x3 + 0, 0) :|: z1 = 0, x >= 0, z = x, z'' = 1 + x3 + 0, x3 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + x3 + 0, 0) :|: xs2 >= 0, z1 = xs2, x >= 0, z = x, z'' = 1 + x3 + 0, x3 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + x3 + 0, 1 + x7 + 0) :|: x7 >= 0, x >= 0, z = x, z'' = 1 + x3 + 0, z1 = 1 + x7 + 0, x3 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(x', xs, 1 + x + xs1, xs2) :|: z = 2, z' = x', xs >= 0, xs2 >= 0, z2 = xs2, x' >= 0, x >= 0, z1 = xs1, xs1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(x', x) + x' + (1 + x + xs) + xs1 + xs2 :|: z' = x', xs >= 0, xs2 >= 0, z = 1, z2 = xs2, x' >= 0, x >= 0, z1 = xs1, xs1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: 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


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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0


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

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(29) 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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(31) 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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(35) 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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(37) 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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](gr(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {lt}, {app}, {part[Ite][True][Ite],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]

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {lt}, {app}, {part[Ite][True][Ite],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]

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

(41) 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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {lt}, {app}, {part[Ite][True][Ite],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]

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

(43) 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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + lt(z', x) + z' + (1 + x + xs) + z1 + z2 :|: xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {part[Ite][True][Ite],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]

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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 }-> s'' :|: s'' >= 0, s'' <= 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + s1 + z' + (1 + x + xs) + z1 + z2 :|: s1 >= 0, s1 <= 2, xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {part[Ite][True][Ite],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]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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 }-> s'' :|: s'' >= 0, s'' <= 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + s1 + z' + (1 + x + xs) + z1 + z2 :|: s1 >= 0, s1 <= 2, xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {part[Ite][True][Ite],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']

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: 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 }-> s'' :|: s'' >= 0, s'' <= 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', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 0) :|: z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 0) :|: z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> app(0, 0) :|: z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, 1 + (z1 - 1) + 0) :|: z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, 1 + (z1 - 1) + 0) :|: z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(1 + (z'' - 1) + 0, 0) :|: z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + s1 + z' + (1 + x + xs) + z1 + z2 :|: s1 >= 0, s1 <= 2, xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part[Ite][True][Ite],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']

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 2 + xs }-> 1 + x + s11 :|: s11 >= 0, s11 <= 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 }-> s'' :|: s'' >= 0, s'' <= 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', z'', z1) -{ 2 }-> s10 :|: s10 >= 0, s10 <= 0 + 0, z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 + z'' }-> s2 :|: s2 >= 0, s2 <= 1 + (z'' - 1) + 0 + (1 + (z1 - 1) + 0), z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 + z'' }-> s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1) + 0 + 0, z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z'' - 1) + 0 + 0, z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + (1 + (z1 - 1) + 0), z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + (z1 - 1) + 0), z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + s1 + z' + (1 + x + xs) + z1 + z2 :|: s1 >= 0, s1 <= 2, xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part[Ite][True][Ite],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']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: part[Ite][True][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 2 + xs }-> 1 + x + s11 :|: s11 >= 0, s11 <= 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 }-> s'' :|: s'' >= 0, s'' <= 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', z'', z1) -{ 2 }-> s10 :|: s10 >= 0, s10 <= 0 + 0, z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 + z'' }-> s2 :|: s2 >= 0, s2 <= 1 + (z'' - 1) + 0 + (1 + (z1 - 1) + 0), z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 + z'' }-> s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1) + 0 + 0, z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z'' - 1) + 0 + 0, z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + (1 + (z1 - 1) + 0), z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + (z1 - 1) + 0), z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + s1 + z' + (1 + x + xs) + z1 + z2 :|: s1 >= 0, s1 <= 2, xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part[Ite][True][Ite],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']
  part[Ite][True][Ite]: runtime: ?, size: INF
  part: runtime: ?, size: INF

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 2 + xs }-> 1 + x + s11 :|: s11 >= 0, s11 <= 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 }-> s'' :|: s'' >= 0, s'' <= 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', z'', z1) -{ 2 }-> s10 :|: s10 >= 0, s10 <= 0 + 0, z1 >= 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 + z'' }-> s2 :|: s2 >= 0, s2 <= 1 + (z'' - 1) + 0 + (1 + (z1 - 1) + 0), z1 - 1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 + z'' }-> s3 :|: s3 >= 0, s3 <= 1 + (z'' - 1) + 0 + 0, z1 = 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z'' - 1) + 0 + 0, z1 >= 0, z >= 0, z'' - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + (1 + (z1 - 1) + 0), z'' = 0, z >= 0, z1 - 1 >= 0, z' = 0
   part(z, z', z'', z1) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z'' = 0, z1 = 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z'' = 0, z1 >= 0, z >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + (z1 - 1) + 0), z >= 0, z1 - 1 >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z1 = 0, z >= 0, z'' >= 0, z' = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](s, 1 + (z - 1), 1 + (1 + y') + xs, z'', z1) :|: s >= 0, s <= 2, xs >= 0, z1 >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z'' >= 0, z - 1 >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z - 1 >= 0, z1 >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](1, 0, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   part(z, z', z'', z1) -{ 1 }-> part[Ite][True][Ite](0, z, 1 + x + xs, z'', z1) :|: xs >= 0, z1 >= 0, z' = 1 + x + xs, z >= 0, x >= 0, z'' >= 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), part(x4, 1 + x4 + (1 + x'1 + xs''), 1 + x4 + 0, 0)) :|: x4 >= 0, z1 = 1 + x4 + (1 + x'1 + xs''), z'' = 1 + x2 + (1 + x''' + xs'), x'1 >= 0, z >= 0, xs' >= 0, xs'' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 = 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 0) :|: z1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(part(x2, 1 + x2 + (1 + x''' + xs'), 1 + x2 + 0, 0), 1 + (z1 - 1) + 0) :|: z1 - 1 >= 0, z'' = 1 + x2 + (1 + x''' + xs'), z >= 0, xs' >= 0, x''' >= 0, x2 >= 0, z' = 0
   part(z, z', z'', z1) -{ 2 }-> app(0, part(x10, 1 + x10 + (1 + x'4 + xs5), 1 + x10 + 0, 0)) :|: z1 = 1 + x10 + (1 + x'4 + xs5), z >= 0, z'' >= 0, x10 >= 0, x'4 >= 0, xs5 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(0, part(x8, 1 + x8 + (1 + x'3 + xs4), 1 + x8 + 0, 0)) :|: z'' = 0, x8 >= 0, x'3 >= 0, z >= 0, z1 = 1 + x8 + (1 + x'3 + xs4), xs4 >= 0, z' = 0
   part(z, z', z'', z1) -{ 3 }-> app(1 + (z'' - 1) + 0, part(x6, 1 + x6 + (1 + x'2 + xs3), 1 + x6 + 0, 0)) :|: z >= 0, x6 >= 0, x'2 >= 0, z1 = 1 + x6 + (1 + x'2 + xs3), z'' - 1 >= 0, z' = 0, xs3 >= 0
   part(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> part(z', xs, 1 + x + z1, z2) :|: z = 2, xs >= 0, z2 >= 0, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   part[Ite][True][Ite](z, z', z'', z1, z2) -{ 0 }-> 1 + s1 + z' + (1 + x + xs) + z1 + z2 :|: s1 >= 0, s1 <= 2, xs >= 0, z2 >= 0, z = 1, z' >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs
   quicksort(z) -{ 1 }-> part(x, 1 + x + (1 + x' + xs), 1 + x + 0, 0) :|: xs >= 0, x >= 0, x' >= 0, z = 1 + x + (1 + x' + xs)
   quicksort(z) -{ 1 }-> 0 :|: z = 0
   quicksort(z) -{ 0 }-> 0 :|: z >= 0
   quicksort(z) -{ 1 }-> 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {part[Ite][True][Ite],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']
  part[Ite][True][Ite]: runtime: INF, size: INF
  part: runtime: ?, size: INF

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

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

(58)
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
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   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
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   QUICKSORT(Cons(z0, Nil)) -> c9
   QUICKSORT(Nil) -> c10
   PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   APP(Nil, z0) -> c15
   NOTEMPTY(Cons(z0, z1)) -> c16
   NOTEMPTY(Nil) -> c17
   GOAL(z0) -> c18(QUICKSORT(z0))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   QUICKSORT(Cons(z0, Nil)) -> c9
   QUICKSORT(Nil) -> c10
   PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   APP(Nil, z0) -> c15
   NOTEMPTY(Cons(z0, z1)) -> c16
   NOTEMPTY(Nil) -> c17
   GOAL(z0) -> c18(QUICKSORT(z0))
K tuples:none
Defined Rule Symbols:   quicksort_1, part_4, app_2, notEmpty_1, goal_1, <_2, >_2, part[Ite][True][Ite]_5

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_1, c1, c2, c3_1, c4, c5, c6_1, c7_1, c8_1, c9, c10, c11_2, c12_2, c13_2, c14_1, c15, c16, c17, c18_1


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

(59) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c18(QUICKSORT(z0))
Removed 9 trailing nodes:
   NOTEMPTY(Nil) -> c17
   <'(0, S(z0)) -> c1
   QUICKSORT(Cons(z0, Nil)) -> c9
   >'(S(z0), 0) -> c5
   <'(z0, 0) -> c2
   >'(0, z0) -> c4
   QUICKSORT(Nil) -> c10
   NOTEMPTY(Cons(z0, z1)) -> c16
   APP(Nil, z0) -> c15

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

(60)
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
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> quicksort(z0)
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
K tuples:none
Defined Rule Symbols:   quicksort_1, part_4, app_2, notEmpty_1, goal_1, <_2, >_2, part[Ite][True][Ite]_5

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c11_2, c12_2, c13_2, c14_1


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

(61) 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)

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
K tuples:none
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c11_2, c12_2, c13_2, c14_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Cons(z1, z2), z3, z4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), z0, Cons(z1, z2), z3, z4), >'(z0, z1)) by 
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4), >'(0, z0))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4), >'(S(z0), 0))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4), >'(0, z0))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4), >'(S(z0), 0))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4), >'(0, z0))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4), >'(S(z0), 0))
K tuples:none
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c12_2, c13_2, c14_1, c11_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
K tuples:none
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c12_2, c13_2, c14_1, c11_2, c11_1


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

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   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)) = 0
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = [1] + x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_1 + x_5
   POL(QUICKSORT(x_1)) = [1]
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = x_1

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c12_2, c13_2, c14_1, c11_2, c11_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil, z1, z2) -> c12(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z1)) by 
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, x2) -> c12(APP(Nil, quicksort(x2)), QUICKSORT(Nil))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, x2) -> c12(APP(Nil, quicksort(x2)), QUICKSORT(Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, x2) -> c12(APP(Nil, quicksort(x2)), QUICKSORT(Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c13_2, c14_1, c11_2, c11_1, c12_2


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil, Nil, x2) -> c12(APP(Nil, quicksort(x2)), QUICKSORT(Nil))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(Cons(z0, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(Cons(z0, Nil)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c13_2, c14_1, c11_2, c11_1, c12_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c13_2, c14_1, c11_2, c11_1, c12_2, c12_1


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

(75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   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]
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5
   POL(QUICKSORT(x_1)) = 0
   POL(S(x_1)) = 0
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c13_2, c14_1, c11_2, c11_1, c12_2, c12_1


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

(77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1]
   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)) = x_1
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = [1]
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2
   POL(QUICKSORT(x_1)) = [1]
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, PART_4, APP_2

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c13_2, c14_1, c11_2, c11_1, c12_2, c12_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(z0, Nil, z1, z2) -> c13(APP(quicksort(z1), quicksort(z2)), QUICKSORT(z2)) by 
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil), QUICKSORT(Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Nil, x2) -> c13(APP(Nil, quicksort(x2)), QUICKSORT(x2))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil), QUICKSORT(Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Nil, x2) -> c13(APP(Nil, quicksort(x2)), QUICKSORT(x2))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil), QUICKSORT(Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Nil, x2) -> c13(APP(Nil, quicksort(x2)), QUICKSORT(x2))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_2, c11_1, c12_2, c12_1, c13_2


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

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(x0, Nil, Nil, x2) -> c13(QUICKSORT(x2))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(x0, Nil, Nil, x2) -> c13(QUICKSORT(x2))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1


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

(83) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PART(x0, Nil, Nil, x2) -> c13(QUICKSORT(x2))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1


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

(85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   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]
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5
   POL(QUICKSORT(x_1)) = 0
   POL(S(x_1)) = 0
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [1] + x_1 + x_2
   POL(<'(x_1, x_2)) = [1]
   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]
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = [1]
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(QUICKSORT(x_1)) = [1]
   POL(S(x_1)) = x_1
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1)))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(S(z0), Cons(S(z1), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(z0), Cons(S(z1), x2), x3, x4), >'(S(z0), S(z1))) by 
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4), >'(S(0), S(z0)))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4), >'(S(0), S(z0)))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4), >'(S(0), S(z0)))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   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)) = 0
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = [1]
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2
   POL(QUICKSORT(x_1)) = [1]
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = x_1

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1)) by 
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c12(APP(Nil, part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Nil))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c12(APP(Nil, part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c12(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c12(APP(Nil, part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Nil))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1)) by 
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Cons(x2, Nil)) -> c12(APP(Nil, Cons(x2, Nil)), QUICKSORT(Nil))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Cons(x2, Nil)) -> c12(APP(Nil, Cons(x2, Nil)), QUICKSORT(Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c12(APP(quicksort(x1), Cons(z0, Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil, Nil, Cons(x2, Nil)) -> c12(APP(Nil, Cons(x2, Nil)), QUICKSORT(Nil))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, x1, Nil) -> c12(APP(quicksort(x1), Nil), QUICKSORT(x1)) by 
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Nil) -> c12(APP(Nil, Nil), QUICKSORT(Nil))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Nil) -> c12(APP(Nil, Nil), QUICKSORT(Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Nil, Nil) -> c12(APP(Nil, Nil), QUICKSORT(Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil, Nil, Nil) -> c12(APP(Nil, Nil), QUICKSORT(Nil))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil), QUICKSORT(Cons(z0, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil), QUICKSORT(Cons(z0, Nil)))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [3] + [3]x_1 + [3]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)) = 0
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = [1]
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2
   POL(QUICKSORT(x_1)) = [1]
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(app(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = 0
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [3] + [2]x_2 + [3]x_3 + [3]x_4 + [3]x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [3] + x_2 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_2, c12_1, c13_2, c13_1, c11_2, c1_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(Cons(z0, Cons(z1, z2)))) by 
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(117) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   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]
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_5
   POL(QUICKSORT(x_1)) = 0
   POL(S(x_1)) = 0
   POL(True) = 0
   POL(app(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = 0
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = 0
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [3] + x_2 + x_4 + x_5
   POL(quicksort(x_1)) = [3]x_1

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2)))) by 
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c13(APP(Nil, part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c13(APP(Nil, part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Cons(z1, z2))) -> c13(APP(quicksort(x1), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(123) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PART(x0, Nil, Nil, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), x2) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), quicksort(x2)), QUICKSORT(x2)) by 
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil), QUICKSORT(Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil), QUICKSORT(Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)), QUICKSORT(Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil), QUICKSORT(Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1]
   POL(<'(x_1, x_2)) = 0
   POL(>(x_1, x_2)) = x_1 + x_2
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1]
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_5
   POL(QUICKSORT(x_1)) = 0
   POL(S(x_1)) = [1]
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(131) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1]
   POL(<'(x_1, x_2)) = [1]
   POL(>(x_1, x_2)) = x_1 + x_2
   POL(>'(x_1, x_2)) = 0
   POL(APP(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = 0
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = [1]
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = [1] + x_3
   POL(QUICKSORT(x_1)) = [1]
   POL(S(x_1)) = [1]
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil))) by 
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Nil, Cons(x2, Nil)) -> c13(APP(Nil, Cons(x2, Nil)))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Nil, Cons(x2, Nil)) -> c13(APP(Nil, Cons(x2, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Cons(z0, Nil)) -> c13(APP(quicksort(x1), Cons(z0, Nil)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil, Nil, Cons(x2, Nil)) -> c13(APP(Nil, Cons(x2, Nil)))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c13_1, c11_2, c1_1, c12_2


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil)) by 
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Nil, Nil) -> c13(APP(Nil, Nil))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Nil, Nil) -> c13(APP(Nil, Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, x1, Nil) -> c13(APP(quicksort(x1), Nil))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c11_2, c1_1, c12_2, c13_1


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

(139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(x0, Nil, Nil, Nil) -> c13(APP(Nil, Nil))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c6_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c11_2, c1_1, c12_2, c13_1


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

(141) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3, z4) -> c6(PART(z0, z2, Cons(z1, z3), z4)) by 
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, PART[ITE][TRUE][ITE]_5, QUICKSORT_1, APP_2, PART_4

Compound Symbols:   c_1, c3_1, c7_1, c8_1, c14_1, c11_1, c12_1, c13_2, c11_2, c1_1, c12_2, c13_1, c6_1


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

(143) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3, z4) -> c7(<'(z0, z1)) by 
   PART[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2, x3) -> c7(<'(0, x0))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2, x3) -> c7(<'(0, x0))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5

Compound Symbols:   c_1, c3_1, c8_1, c14_1, c11_1, c12_1, c13_2, c11_2, c1_1, c12_2, c13_1, c6_1, c7_1


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

(145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   PART[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2, x3) -> c7(<'(0, x0))
   PART(0, Cons(z0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3, x4))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5

Compound Symbols:   c_1, c3_1, c8_1, c14_1, c11_1, c12_1, c13_2, c11_2, c1_1, c12_2, c13_1, c6_1, c7_1


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

(147) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(S(z0), Cons(0, x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3, x4)) by 
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5

Compound Symbols:   c_1, c3_1, c8_1, c14_1, c12_1, c13_2, c11_2, c11_1, c1_1, c12_2, c13_1, c6_1, c7_1


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

(149) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(z0, Nil), x2) -> c12(APP(Cons(z0, Nil), quicksort(x2))) by 
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5

Compound Symbols:   c_1, c3_1, c8_1, c14_1, c13_2, c11_2, c11_1, c1_1, c12_2, c12_1, c13_1, c6_1, c7_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5

Compound Symbols:   c_1, c3_1, c8_1, c14_1, c13_2, c11_2, c11_1, c1_1, c12_2, c12_1, c13_1, c6_1, c7_1


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

(153) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(z0, Nil), x2) -> c13(APP(Cons(z0, Nil), quicksort(x2)), QUICKSORT(x2)) by 
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   <'(S(z0), S(z1)) -> c(<'(z0, z1))
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   <'_2, >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5

Compound Symbols:   c_1, c3_1, c8_1, c14_1, c11_2, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1


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

(155) 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)))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_2, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1


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

(157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(PART[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3, x4))
   PART[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2, x3) -> c7(<'(S(0), S(x0)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_2, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1


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

(159) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(S(S(z0)), Cons(S(S(z1)), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), x2), x3, x4), >'(S(S(z0)), S(S(z1)))) by 
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_2, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c12(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil))) by PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_2, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1


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

(163) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(S(S(z0)), Cons(S(0), x2), x3, x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3, x4), >'(S(S(z0)), S(0))) by 
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1, c11_2


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

(165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
We considered the (Usable) Rules:none
And the Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1]
   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)) = x_1
   POL(False) = [1]
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = [1]
   POL(True) = 0
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = x_1
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = x_1

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1, c11_2


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

(167) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
And the Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   POL(<'(x_1, x_2)) = x_1 + x_2
   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]
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_5
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = x_1
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = x_1

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1, c11_2


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1, c11_2


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

(171) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1)))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c11_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c_1, c11_2


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

(173) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(S(x0), Cons(S(x1), x2), x3, x4) -> c11(>'(S(x0), S(x1))) by 
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2


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

(175) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(x1, Cons(x2, x3)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2


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

(177) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)), QUICKSORT(Cons(x1, Cons(x2, x3)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c12(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil), QUICKSORT(Cons(x1, Cons(x2, x3)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c1_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2


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

(181) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(S(0), Cons(S(z0), x2), x3, x4) -> c1(>'(S(0), S(z0))) by 
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(183) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(185) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c12(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(x1)) by 
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_1, c12_2, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Nil), Cons(x2, Cons(x3, x4))) -> c13(APP(Cons(z0, Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4)))) by PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_1, c12_2, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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(189) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c12(APP(Cons(z0, Nil), Cons(x2, Nil))) by 
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_1, c12_2, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(191) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Cons(z1, z2))) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2)))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_1, c12_2, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_1, c12_2, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(195) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(z0, Nil), Nil) -> c12(APP(Cons(z0, Nil), Nil)) by 
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   >'(S(z0), S(z1)) -> c3(>'(z0, z1))
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   >'_2, QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2

Compound Symbols:   c3_1, c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1


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

(197) 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)))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c12(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)), QUICKSORT(Cons(z0, Cons(z1, z2))))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Cons(x3, x4))) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), part(x2, Cons(x2, Cons(x3, x4)), Cons(x2, Nil), Nil)), QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c1_1, c3_1


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

(199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   PART(S(0), Cons(S(z0), z1), Cons(0, x2), x3) -> c1(>'(S(0), S(z0)))
   PART(S(0), Cons(S(0), Cons(x1, x2)), Cons(S(0), Nil), Nil) -> c1(>'(S(0), S(0)))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3), >'(S(S(z0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(0)))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3), >'(S(S(x0)), S(0)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c3_1


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

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c3_1


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

(203) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil)) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c3_1


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

(205) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), Cons(x2, Nil)) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Cons(x2, Nil))) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c3_1


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

(207) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PART(x0, Nil, Cons(z0, Cons(z1, z2)), Nil) -> c13(APP(part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil), Nil)) by PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
----------------------------------------

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3))))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_2, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c_1, c11_2, c3_1


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

(209) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c12(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(Cons(z0, Cons(x2, x3)))) by 
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c12_1, c13_2, c13_1, c6_1, c7_1, c11_1, c12_2, c_1, c11_2, c3_1


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

(211) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(x1, Cons(x2, x3)), x4) -> c12(QUICKSORT(Cons(x1, Cons(x2, x3)))) by 
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
S tuples:
   QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil))
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   QUICKSORT_1, APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2

Compound Symbols:   c8_1, c14_1, c13_2, c13_1, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1


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

(213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace QUICKSORT(Cons(z0, Cons(z1, z2))) -> c8(PART(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)) by 
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(0, x2)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(0, Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_2, c13_1, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1, c8_1


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

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_2, c13_1, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1, c8_1, c12


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(217) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
We considered the (Usable) Rules:
   >(S(z0), 0) -> True
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
And the Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(<(x_1, x_2)) = [1]
   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)) = x_1
   POL(False) = 0
   POL(Nil) = 0
   POL(PART(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(PART[ITE][TRUE][ITE](x_1, x_2, x_3, x_4, x_5)) = x_1 + x_5
   POL(QUICKSORT(x_1)) = x_1
   POL(S(x_1)) = [1]
   POL(True) = [1]
   POL(app(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12) = 0
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   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(part(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_4
   POL(part[Ite][True][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 + x_4 + x_5
   POL(part[Ite][True][Ite][False][Ite](x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4 + x_5
   POL(quicksort(x_1)) = 0

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4))))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_2, c13_1, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1, c8_1, c12


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

(219) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, x1, Cons(z0, Cons(x3, x4))) -> c13(APP(quicksort(x1), part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x3, x4)), Cons(z0, Nil), Nil)), QUICKSORT(Cons(z0, Cons(x3, x4)))) by 
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
K tuples:
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Cons(z0, Nil)) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Cons(z0, Nil)))
   PART(x0, Nil, Cons(x1, Cons(x2, x3)), Nil) -> c13(APP(part(x1, Cons(x1, Cons(x2, x3)), Cons(x1, Nil), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_1, c13_2, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1, c8_1, c12


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

(221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4))))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
K tuples:
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_1, c13_2, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1, c8_1


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(223) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, x1, Cons(x2, Cons(x3, x4))) -> c13(QUICKSORT(Cons(x2, Cons(x3, x4)))) by 
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
K tuples:
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_2, c13_1, c6_1, c7_1, c11_1, c12_1, c12_2, c_1, c11_2, c3_1, c8_1


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

(225) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PART(x0, Nil, Cons(z0, Cons(x2, x3)), x4) -> c13(APP(part[Ite][True][Ite](>(z0, z0), z0, Cons(z0, Cons(x2, x3)), Cons(z0, Nil), Nil), quicksort(x4)), QUICKSORT(x4)) by 
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c13(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c13(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c13(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(x3))

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

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   >(S(z0), S(z1)) -> >(z0, z1)
   >(0, z0) -> False
   >(S(z0), 0) -> True
   quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z0, Cons(z1, z2)), Cons(z0, Nil), Nil)
   quicksort(Cons(z0, Nil)) -> Cons(z0, Nil)
   quicksort(Nil) -> Nil
   part(z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite](>(z0, z1), z0, Cons(z1, z2), z3, z4)
   part(z0, Nil, z1, z2) -> app(quicksort(z1), quicksort(z2))
   part[Ite][True][Ite](True, z0, Cons(z1, z2), z3, z4) -> part(z0, z2, Cons(z1, z3), z4)
   part[Ite][True][Ite](False, z0, Cons(z1, z2), z3, z4) -> part[Ite][True][Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2), z3, z4)
   app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2))
   app(Nil, z0) -> z0
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
Tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(x0, Nil, Cons(z0, Nil), Cons(x2, Nil)) -> c13(APP(Cons(z0, Nil), Cons(x2, Nil)))
   PART(x0, Nil, Cons(z0, Nil), Nil) -> c13(APP(Cons(z0, Nil), Nil))
   PART[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2, x3) -> c6(PART(S(x0), x1, Cons(0, x2), x3))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c6(PART(S(S(x0)), x2, Cons(S(S(x1)), x3), x4))
   PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2, x3) -> c6(PART(S(S(x0)), x1, Cons(S(0), x2), x3))
   PART[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3, x4) -> c7(<'(S(S(x0)), S(S(x1))))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c12(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   <'(S(S(y0)), S(S(y1))) -> c(<'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c12(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6))) -> c13(APP(part(z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), part[Ite][True][Ite](>(z4, z4), z4, Cons(z4, Cons(z5, z6)), Cons(z4, Nil), Nil)), QUICKSORT(Cons(z4, Cons(z5, z6))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(S(x1)), x3)))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(S(0), x2)))
   PART(z0, Nil, Cons(z1, Nil), Cons(z2, Cons(z3, z4))) -> c13(APP(Cons(z1, Nil), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Nil), Cons(z2, Nil)) -> c12(APP(Cons(0, Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(S(x1)), Nil), Cons(z2, Nil)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Cons(z2, Nil)) -> c12(APP(Cons(S(0), Nil), Cons(z2, Nil)))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Cons(z4, Nil)) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Cons(z4, Nil)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   >'(S(S(y0)), S(S(y1))) -> c3(>'(S(y0), S(y1)))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c13(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c12(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(0, x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(S(x1)), x3)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(APP(quicksort(Cons(S(0), x2)), part[Ite][True][Ite](>(z2, z2), z2, Cons(z2, Cons(z3, z4)), Cons(z2, Nil), Nil)), QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, x2), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), x3), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(S(x0)), Nil, Cons(S(0), x2), Cons(z2, Cons(z3, z4))) -> c13(QUICKSORT(Cons(z2, Cons(z3, z4))))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c13(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c13(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c13(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(x3))
S tuples:
   APP(Cons(z0, z1), z2) -> c14(APP(z1, z2))
   PART(S(x0), Cons(0, z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(x0), Cons(0, z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, z1), Cons(S(0), x2), x3))
   PART(S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil) -> c11(PART[ITE][TRUE][ITE](>(z0, z0), S(S(z0)), Cons(S(S(z0)), Cons(x1, x2)), Cons(S(S(z0)), Nil), Nil), >'(S(S(z0)), S(S(z0))))
   PART(S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](>(z0, z1), S(S(z0)), Cons(S(S(z1)), z2), Cons(0, x2), x3), >'(S(S(z0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(S(x1)), x3), x4), >'(S(S(x0)), S(S(z1))))
   PART(S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](>(x0, z1), S(S(x0)), Cons(S(S(z1)), z2), Cons(S(0), x2), x3), >'(S(S(x0)), S(S(z1))))
   PART(z0, Nil, Cons(z1, Cons(z2, z3)), Nil) -> c12(APP(part[Ite][True][Ite](>(z1, z1), z1, Cons(z1, Cons(z2, z3)), Cons(z1, Nil), Nil), Nil), QUICKSORT(Cons(z1, Cons(z2, z3))))
   PART(S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), z1), Cons(0, x2), x3))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(S(x1)), x3), x4))
   PART(S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3) -> c11(PART[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), z1), Cons(S(0), x2), x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(Cons(S(0), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c12(QUICKSORT(Cons(S(S(x1)), Cons(z2, z3))))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c12(QUICKSORT(Cons(S(0), Cons(z2, z3))))
   QUICKSORT(Cons(S(S(y0)), Cons(z1, z2))) -> c8(PART(S(S(y0)), Cons(S(S(y0)), Cons(z1, z2)), Cons(S(S(y0)), Nil), Nil))
   QUICKSORT(Cons(S(y0), Cons(z1, z2))) -> c8(PART(S(y0), Cons(S(y0), Cons(z1, z2)), Cons(S(y0), Nil), Nil))
K tuples:
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c12(APP(Cons(0, Nil), quicksort(x3)))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c12(APP(Cons(S(S(x1)), Nil), quicksort(x4)))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c12(APP(Cons(S(0), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Nil), x3) -> c13(APP(Cons(0, Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), x4) -> c13(APP(Cons(S(S(x1)), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), x3) -> c13(APP(Cons(S(0), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(z0), Cons(S(z0), Cons(x1, x2)), Cons(S(z0), Nil), Nil) -> c11(>'(S(z0), S(z0)))
   PART(S(x0), Cons(S(z1), z2), Cons(0, x2), x3) -> c11(>'(S(x0), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(S(x1)), x3), x4) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(S(x0)), Cons(S(z1), z2), Cons(S(0), x2), x3) -> c11(>'(S(S(x0)), S(z1)))
   PART(S(x0), Nil, Cons(0, Nil), Nil) -> c12(APP(Cons(0, Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Nil), Nil) -> c12(APP(Cons(S(S(x1)), Nil), Nil))
   PART(S(S(x0)), Nil, Cons(S(0), Nil), Nil) -> c12(APP(Cons(S(0), Nil), Nil))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c12(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)))
   PART(S(x0), Nil, Cons(0, Cons(z2, z3)), x3) -> c13(APP(part[Ite][True][Ite](>(0, 0), 0, Cons(0, Cons(z2, z3)), Cons(0, Nil), Nil), quicksort(x3)), QUICKSORT(x3))
   PART(S(S(x0)), Nil, Cons(S(S(x1)), Cons(z2, z3)), x4) -> c13(APP(part[Ite][True][Ite](>(S(S(x1)), S(S(x1))), S(S(x1)), Cons(S(S(x1)), Cons(z2, z3)), Cons(S(S(x1)), Nil), Nil), quicksort(x4)), QUICKSORT(x4))
   PART(S(S(x0)), Nil, Cons(S(0), Cons(z2, z3)), x3) -> c13(APP(part[Ite][True][Ite](>(S(0), S(0)), S(0), Cons(S(0), Cons(z2, z3)), Cons(S(0), Nil), Nil), quicksort(x3)), QUICKSORT(x3))
Defined Rule Symbols:   >_2, quicksort_1, part_4, part[Ite][True][Ite]_5, app_2, <_2

Defined Pair Symbols:   APP_2, PART_4, PART[ITE][TRUE][ITE]_5, <'_2, >'_2, QUICKSORT_1

Compound Symbols:   c14_1, c13_1, c6_1, c7_1, c11_1, c12_1, c12_2, c13_2, c_1, c11_2, c3_1, c8_1