KILLED
proof of input_rap1UPs9Y7.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 217 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) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 16 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 21 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 459 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 118 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 323 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 2025 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 631 ms]
        (46) CpxRNTS
        (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 7945 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 2182 ms]
        (52) CpxRNTS
    (53) CompletionProof [UPPER BOUND(ID), 0 ms]
    (54) CpxTypedWeightedCompleteTrs
        (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
(57) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms]
    (90) CdtProblem
    (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) 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:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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

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

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


The TRS R consists of the following rules:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))
   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs))
   minsort(Nil) -> Nil
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs')
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs)))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs))

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0', S(y)) -> False
   !EQ(S(x), 0') -> False
   !EQ(0', 0') -> True
   <(S(x), S(y)) -> <(x, y)
   <(0', S(y)) -> True
   <(x, 0') -> False
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs))
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs')
   remove[Ite](True, x', Cons(x, xs)) -> xs
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs')

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:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs)) [1]
   minsort(Nil) -> Nil [1]
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](<(x, min), min, Cons(x, xs), xs') [1]
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs))) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs)) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   <(S(x), S(y)) -> <(x, y) [0]
   <(0, S(y)) -> True [0]
   <(x, 0) -> False [0]
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs)) [0]
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs') [0]
   remove[Ite](True, x', Cons(x, xs)) -> xs [0]
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs') [0]

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

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

< => 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:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs)) [1]
   minsort(Nil) -> Nil [1]
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](lt(x, min), min, Cons(x, xs), xs') [1]
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs))) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs)) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs)) [0]
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs') [0]
   remove[Ite](True, x', Cons(x, xs)) -> xs [0]
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs') [0]

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

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

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

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs)) [1]
   minsort(Nil) -> Nil [1]
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](lt(x, min), min, Cons(x, xs), xs') [1]
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs))) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs)) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs)) [0]
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs') [0]
   remove[Ite](True, x', Cons(x, xs)) -> xs [0]
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs') [0]

The TRS has the following type information:
minsort :: Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
appmin :: S:0 -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
appmin[Ite][True][Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False
remove :: S:0 -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
remove[Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil
!EQ :: S:0 -> S:0 -> True:False
S :: S:0 -> S:0
0 :: S:0

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

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

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

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   minsort_1
   appmin_3
   notEmpty_1

(c) The following functions are completely defined:

   remove_2
   !EQ_2
   lt_2
   remove[Ite]_3
   appmin[Ite][True][Ite]_4

Due to the following rules being added:

   !EQ(v0, v1) -> null_!EQ [0]
   lt(v0, v1) -> null_lt [0]
   remove[Ite](v0, v1, v2) -> Nil [0]
   appmin[Ite][True][Ite](v0, v1, v2, v3) -> Nil [0]
   remove(v0, v1) -> Nil [0]

And the following fresh constants:    null_!EQ, null_lt

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

(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs)) [1]
   minsort(Nil) -> Nil [1]
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](lt(x, min), min, Cons(x, xs), xs') [1]
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs))) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs)) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs)) [0]
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs') [0]
   remove[Ite](True, x', Cons(x, xs)) -> xs [0]
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs') [0]
   !EQ(v0, v1) -> null_!EQ [0]
   lt(v0, v1) -> null_lt [0]
   remove[Ite](v0, v1, v2) -> Nil [0]
   appmin[Ite][True][Ite](v0, v1, v2, v3) -> Nil [0]
   remove(v0, v1) -> Nil [0]

The TRS has the following type information:
minsort :: Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
appmin :: S:0 -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
appmin[Ite][True][Ite] :: True:False:null_!EQ:null_lt -> S:0 -> Cons:Nil -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False:null_!EQ:null_lt
remove :: S:0 -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False:null_!EQ:null_lt
True :: True:False:null_!EQ:null_lt
False :: True:False:null_!EQ:null_lt
remove[Ite] :: True:False:null_!EQ:null_lt -> S:0 -> Cons:Nil -> Cons:Nil
!EQ :: S:0 -> S:0 -> True:False:null_!EQ:null_lt
S :: S:0 -> S:0
0 :: S:0
null_!EQ :: True:False:null_!EQ:null_lt
null_lt :: True:False:null_!EQ:null_lt

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs)) [1]
   minsort(Nil) -> Nil [1]
   appmin(S(y'), Cons(S(x''), xs), xs') -> appmin[Ite][True][Ite](lt(x'', y'), S(y'), Cons(S(x''), xs), xs') [1]
   appmin(S(y''), Cons(0, xs), xs') -> appmin[Ite][True][Ite](True, S(y''), Cons(0, xs), xs') [1]
   appmin(0, Cons(x, xs), xs') -> appmin[Ite][True][Ite](False, 0, Cons(x, xs), xs') [1]
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](null_lt, min, Cons(x, xs), xs') [1]
   appmin(min, Nil, Cons(x1, xs'')) -> Cons(min, minsort(remove[Ite](!EQ(min, x1), min, Cons(x1, xs'')))) [2]
   appmin(min, Nil, xs) -> Cons(min, minsort(Nil)) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   remove(S(x2), Cons(S(y1), xs)) -> remove[Ite](!EQ(x2, y1), S(x2), Cons(S(y1), xs)) [1]
   remove(0, Cons(S(y2), xs)) -> remove[Ite](False, 0, Cons(S(y2), xs)) [1]
   remove(S(x3), Cons(0, xs)) -> remove[Ite](False, S(x3), Cons(0, xs)) [1]
   remove(0, Cons(0, xs)) -> remove[Ite](True, 0, Cons(0, xs)) [1]
   remove(x', Cons(x, xs)) -> remove[Ite](null_!EQ, x', Cons(x, xs)) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs)) [0]
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs') [0]
   remove[Ite](True, x', Cons(x, xs)) -> xs [0]
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs') [0]
   !EQ(v0, v1) -> null_!EQ [0]
   lt(v0, v1) -> null_lt [0]
   remove[Ite](v0, v1, v2) -> Nil [0]
   appmin[Ite][True][Ite](v0, v1, v2, v3) -> Nil [0]
   remove(v0, v1) -> Nil [0]

The TRS has the following type information:
minsort :: Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
appmin :: S:0 -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
appmin[Ite][True][Ite] :: True:False:null_!EQ:null_lt -> S:0 -> Cons:Nil -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False:null_!EQ:null_lt
remove :: S:0 -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False:null_!EQ:null_lt
True :: True:False:null_!EQ:null_lt
False :: True:False:null_!EQ:null_lt
remove[Ite] :: True:False:null_!EQ:null_lt -> S:0 -> Cons:Nil -> Cons:Nil
!EQ :: S:0 -> S:0 -> True:False:null_!EQ:null_lt
S :: S:0 -> S:0
0 :: S:0
null_!EQ :: True:False:null_!EQ:null_lt
null_lt :: True:False:null_!EQ:null_lt

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

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

   Nil => 0
   True => 2
   False => 1
   0 => 0
   null_!EQ => 0
   null_lt => 0

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', y'), 1 + y', 1 + (1 + x'') + xs, xs') :|: z'' = xs', xs >= 0, xs' >= 0, z = 1 + y', y' >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + y'', 1 + 0 + xs, xs') :|: z'' = xs', xs >= 0, z = 1 + y'', z' = 1 + 0 + xs, xs' >= 0, y'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, xs') :|: z'' = xs', xs >= 0, z' = 1 + x + xs, x >= 0, xs' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, min, 1 + x + xs, xs') :|: z'' = xs', xs >= 0, z' = 1 + x + xs, x >= 0, xs' >= 0, z = min, min >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + min + minsort(remove[Ite](!EQ(min, x1), min, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z = min, z'' = 1 + x1 + xs'', min >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + min + minsort(0) :|: xs >= 0, z'' = xs, z = min, min >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(min, xs, xs') :|: xs >= 0, z' = min, z = 1, x >= 0, xs' >= 0, z'' = 1 + x + xs, min >= 0, z1 = xs'
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, xs') :|: z = 2, xs >= 0, z' = min, x >= 0, xs' >= 0, z'' = 1 + x + xs, min >= 0, z1 = xs'
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + x3, 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 1 + x3, x3 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(x2, y1), 1 + x2, 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z = 1 + x2, x2 >= 0
   remove(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs


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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs


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

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

   { notEmpty }
   { !EQ }
   { lt }
   { remove[Ite], remove }
   { appmin, minsort, appmin[Ite][True][Ite] }

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {notEmpty}, {!EQ}, {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}

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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {notEmpty}, {!EQ}, {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {notEmpty}, {!EQ}, {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: ?, size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {!EQ}, {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]

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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {!EQ}, {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {!EQ}, {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: ?, size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   !EQ(z, z') -{ 0 }-> !EQ(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](!EQ(z, x1), z, 1 + x1 + xs'')) :|: x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(z - 1, y1), 1 + (z - 1), 1 + (1 + y1) + xs) :|: y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](s, z, 1 + x1 + xs'')) :|: s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](s', 1 + (z - 1), 1 + (1 + y1) + xs) :|: s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](s, z, 1 + x1 + xs'')) :|: s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](s', 1 + (z - 1), 1 + (1 + y1) + xs) :|: s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {lt}, {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: ?, size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x'', z - 1), 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](s, z, 1 + x1 + xs'')) :|: s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](s', 1 + (z - 1), 1 + (1 + y1) + xs) :|: s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: s1 >= 0, s1 <= 2, xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](s, z, 1 + x1 + xs'')) :|: s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](s', 1 + (z - 1), 1 + (1 + y1) + xs) :|: s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: s1 >= 0, s1 <= 2, xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](s, z, 1 + x1 + xs'')) :|: s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](s', 1 + (z - 1), 1 + (1 + y1) + xs) :|: s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {remove[Ite],remove}, {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  remove[Ite]: runtime: ?, size: O(n^1) [z'']
  remove: runtime: ?, size: O(n^1) [z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: remove[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z''

Computed RUNTIME bound using CoFloCo for: remove
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + z'

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: s1 >= 0, s1 <= 2, xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 2 }-> 1 + z + minsort(remove[Ite](s, z, 1 + x1 + xs'')) :|: s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 1 }-> remove[Ite](s', 1 + (z - 1), 1 + (1 + y1) + xs) :|: s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](2, 0, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 0, 1 + (1 + y2) + xs) :|: xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 1 }-> remove[Ite](1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 1 }-> remove[Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  remove[Ite]: runtime: O(n^1) [3 + z''], size: O(n^1) [z'']
  remove: runtime: O(n^1) [4 + z'], size: O(n^1) [z']

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: s1 >= 0, s1 <= 2, xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 6 + x1 + xs'' }-> 1 + z + minsort(s3) :|: s3 >= 0, s3 <= 1 + x1 + xs'', s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 6 + xs + y1 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + y1) + xs, s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 6 + xs + y2 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + y2) + xs, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 4 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 4 + z' }-> s7 :|: s7 >= 0, s7 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z = 0
   remove(z, z') -{ 5 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 4 + xs }-> 1 + x + s9 :|: s9 >= 0, s9 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  remove[Ite]: runtime: O(n^1) [3 + z''], size: O(n^1) [z'']
  remove: runtime: O(n^1) [4 + z'], size: O(n^1) [z']

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: s1 >= 0, s1 <= 2, xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 6 + x1 + xs'' }-> 1 + z + minsort(s3) :|: s3 >= 0, s3 <= 1 + x1 + xs'', s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 6 + xs + y1 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + y1) + xs, s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 6 + xs + y2 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + y2) + xs, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 4 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 4 + z' }-> s7 :|: s7 >= 0, s7 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z = 0
   remove(z, z') -{ 5 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 4 + xs }-> 1 + x + s9 :|: s9 >= 0, s9 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  remove[Ite]: runtime: O(n^1) [3 + z''], size: O(n^1) [z'']
  remove: runtime: O(n^1) [4 + z'], size: O(n^1) [z']
  appmin: runtime: ?, size: INF
  minsort: runtime: ?, size: INF
  appmin[Ite][True][Ite]: runtime: ?, size: INF

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   !EQ(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](s1, 1 + (z - 1), 1 + (1 + x'') + xs, z'') :|: s1 >= 0, s1 <= 2, xs >= 0, z'' >= 0, z - 1 >= 0, z' = 1 + (1 + x'') + xs, x'' >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](2, 1 + (z - 1), 1 + 0 + (z' - 1), z'') :|: z' - 1 >= 0, z'' >= 0, z - 1 >= 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](1, 0, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](0, z, 1 + x + xs, z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z >= 0
   appmin(z, z', z'') -{ 6 + x1 + xs'' }-> 1 + z + minsort(s3) :|: s3 >= 0, s3 <= 1 + x1 + xs'', s >= 0, s <= 2, x1 >= 0, xs'' >= 0, z'' = 1 + x1 + xs'', z >= 0, z' = 0
   appmin(z, z', z'') -{ 1 }-> 1 + z + minsort(0) :|: z'' >= 0, z >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, z1) :|: z = 2, xs >= 0, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(z', xs, z1) :|: xs >= 0, z = 1, x >= 0, z1 >= 0, z'' = 1 + x + xs, z' >= 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   remove(z, z') -{ 6 + xs + y1 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + y1) + xs, s' >= 0, s' <= 2, y1 >= 0, xs >= 0, z' = 1 + (1 + y1) + xs, z - 1 >= 0
   remove(z, z') -{ 6 + xs + y2 }-> s5 :|: s5 >= 0, s5 <= 1 + (1 + y2) + xs, xs >= 0, z' = 1 + (1 + y2) + xs, y2 >= 0, z = 0
   remove(z, z') -{ 4 + z' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z - 1 >= 0
   remove(z, z') -{ 4 + z' }-> s7 :|: s7 >= 0, s7 <= 1 + 0 + (z' - 1), z' - 1 >= 0, z = 0
   remove(z, z') -{ 5 + x + xs }-> s8 :|: s8 >= 0, s8 <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   remove(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   remove[Ite](z, z', z'') -{ 4 + xs }-> 1 + x + s9 :|: s9 >= 0, s9 <= xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {appmin,minsort,appmin[Ite][True][Ite]}
Previous analysis results are:
  notEmpty: runtime: O(1) [1], size: O(1) [2]
  !EQ: runtime: O(1) [0], size: O(1) [2]
  lt: runtime: O(1) [0], size: O(1) [2]
  remove[Ite]: runtime: O(n^1) [3 + z''], size: O(n^1) [z'']
  remove: runtime: O(n^1) [4 + z'], size: O(n^1) [z']
  appmin: runtime: INF, size: INF
  minsort: runtime: ?, size: INF
  appmin[Ite][True][Ite]: runtime: ?, size: INF

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

(53) 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:

   !EQ(v0, v1) -> null_!EQ [0]
   lt(v0, v1) -> null_lt [0]
   remove[Ite](v0, v1, v2) -> null_remove[Ite] [0]
   appmin[Ite][True][Ite](v0, v1, v2, v3) -> null_appmin[Ite][True][Ite] [0]
   minsort(v0) -> null_minsort [0]
   appmin(v0, v1, v2) -> null_appmin [0]
   notEmpty(v0) -> null_notEmpty [0]
   remove(v0, v1) -> null_remove [0]

And the following fresh constants:    null_!EQ, null_lt, null_remove[Ite], null_appmin[Ite][True][Ite], null_minsort, null_appmin, null_notEmpty, null_remove

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

(54)
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:

   minsort(Cons(x, xs)) -> appmin(x, xs, Cons(x, xs)) [1]
   minsort(Nil) -> Nil [1]
   appmin(min, Cons(x, xs), xs') -> appmin[Ite][True][Ite](lt(x, min), min, Cons(x, xs), xs') [1]
   appmin(min, Nil, xs) -> Cons(min, minsort(remove(min, xs))) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   remove(x', Cons(x, xs)) -> remove[Ite](!EQ(x', x), x', Cons(x, xs)) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   remove[Ite](False, x', Cons(x, xs)) -> Cons(x, remove(x', xs)) [0]
   appmin[Ite][True][Ite](True, min, Cons(x, xs), xs') -> appmin(x, xs, xs') [0]
   remove[Ite](True, x', Cons(x, xs)) -> xs [0]
   appmin[Ite][True][Ite](False, min, Cons(x, xs), xs') -> appmin(min, xs, xs') [0]
   !EQ(v0, v1) -> null_!EQ [0]
   lt(v0, v1) -> null_lt [0]
   remove[Ite](v0, v1, v2) -> null_remove[Ite] [0]
   appmin[Ite][True][Ite](v0, v1, v2, v3) -> null_appmin[Ite][True][Ite] [0]
   minsort(v0) -> null_minsort [0]
   appmin(v0, v1, v2) -> null_appmin [0]
   notEmpty(v0) -> null_notEmpty [0]
   remove(v0, v1) -> null_remove [0]

The TRS has the following type information:
minsort :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
Cons :: S:0 -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
appmin :: S:0 -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
Nil :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
appmin[Ite][True][Ite] :: True:False:null_!EQ:null_lt:null_notEmpty -> S:0 -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
lt :: S:0 -> S:0 -> True:False:null_!EQ:null_lt:null_notEmpty
remove :: S:0 -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
notEmpty :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> True:False:null_!EQ:null_lt:null_notEmpty
True :: True:False:null_!EQ:null_lt:null_notEmpty
False :: True:False:null_!EQ:null_lt:null_notEmpty
remove[Ite] :: True:False:null_!EQ:null_lt:null_notEmpty -> S:0 -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove -> Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
!EQ :: S:0 -> S:0 -> True:False:null_!EQ:null_lt:null_notEmpty
S :: S:0 -> S:0
0 :: S:0
null_!EQ :: True:False:null_!EQ:null_lt:null_notEmpty
null_lt :: True:False:null_!EQ:null_lt:null_notEmpty
null_remove[Ite] :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
null_appmin[Ite][True][Ite] :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
null_minsort :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
null_appmin :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove
null_notEmpty :: True:False:null_!EQ:null_lt:null_notEmpty
null_remove :: Cons:Nil:null_remove[Ite]:null_appmin[Ite][True][Ite]:null_minsort:null_appmin:null_remove

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

(55) 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_!EQ => 0
   null_lt => 0
   null_remove[Ite] => 0
   null_appmin[Ite][True][Ite] => 0
   null_minsort => 0
   null_appmin => 0
   null_notEmpty => 0
   null_remove => 0

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   appmin(z, z', z'') -{ 1 }-> appmin[Ite][True][Ite](lt(x, min), min, 1 + x + xs, xs') :|: z'' = xs', xs >= 0, z' = 1 + x + xs, x >= 0, xs' >= 0, z = min, min >= 0
   appmin(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   appmin(z, z', z'') -{ 1 }-> 1 + min + minsort(remove(min, xs)) :|: xs >= 0, z'' = xs, z = min, min >= 0, z' = 0
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(min, xs, xs') :|: xs >= 0, z' = min, z = 1, x >= 0, xs' >= 0, z'' = 1 + x + xs, min >= 0, z1 = xs'
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> appmin(x, xs, xs') :|: z = 2, xs >= 0, z' = min, x >= 0, xs' >= 0, z'' = 1 + x + xs, min >= 0, z1 = xs'
   appmin[Ite][True][Ite](z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   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
   minsort(z) -{ 1 }-> appmin(x, xs, 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   minsort(z) -{ 1 }-> 0 :|: z = 0
   minsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   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
   remove(z, z') -{ 1 }-> remove[Ite](!EQ(x', x), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   remove(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   remove[Ite](z, z', z'') -{ 0 }-> xs :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
   remove[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   remove[Ite](z, z', z'') -{ 0 }-> 1 + x + remove(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs

Only complete derivations are relevant for the runtime complexity.

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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   appmin[Ite][True][Ite](True, z0, Cons(z1, z2), z3) -> appmin(z1, z2, z3)
   appmin[Ite][True][Ite](False, z0, Cons(z1, z2), z3) -> appmin(z0, z2, z3)
   minsort(Cons(z0, z1)) -> appmin(z0, z1, Cons(z0, z1))
   minsort(Nil) -> Nil
   appmin(z0, Cons(z1, z2), z3) -> appmin[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2), z3)
   appmin(z0, Nil, z1) -> Cons(z0, minsort(remove(z0, z1)))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   !EQ'(0, S(z0)) -> c1
   !EQ'(S(z0), 0) -> c2
   !EQ'(0, 0) -> c3
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   <'(0, S(z0)) -> c5
   <'(z0, 0) -> c6
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   REMOVE[ITE](True, z0, Cons(z1, z2)) -> c8
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   MINSORT(Nil) -> c12
   APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   NOTEMPTY(Cons(z0, z1)) -> c15
   NOTEMPTY(Nil) -> c16
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   MINSORT(Nil) -> c12
   APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   NOTEMPTY(Cons(z0, z1)) -> c15
   NOTEMPTY(Nil) -> c16
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
K tuples:none
Defined Rule Symbols:   minsort_1, appmin_3, notEmpty_1, remove_2, !EQ_2, <_2, remove[Ite]_3, appmin[Ite][True][Ite]_4

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, NOTEMPTY_1, REMOVE_2

Compound Symbols:   c_1, c1, c2, c3, c4_1, c5, c6, c7_1, c8, c9_1, c10_1, c11_1, c12, c13_2, c14_2, c15, c16, c17_2


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing nodes:
   <'(0, S(z0)) -> c5
   NOTEMPTY(Nil) -> c16
   NOTEMPTY(Cons(z0, z1)) -> c15
   !EQ'(S(z0), 0) -> c2
   MINSORT(Nil) -> c12
   !EQ'(0, S(z0)) -> c1
   <'(z0, 0) -> c6
   REMOVE[ITE](True, z0, Cons(z1, z2)) -> c8
   !EQ'(0, 0) -> c3

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   appmin[Ite][True][Ite](True, z0, Cons(z1, z2), z3) -> appmin(z1, z2, z3)
   appmin[Ite][True][Ite](False, z0, Cons(z1, z2), z3) -> appmin(z0, z2, z3)
   minsort(Cons(z0, z1)) -> appmin(z0, z1, Cons(z0, z1))
   minsort(Nil) -> Nil
   appmin(z0, Cons(z1, z2), z3) -> appmin[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2), z3)
   appmin(z0, Nil, z1) -> Cons(z0, minsort(remove(z0, z1)))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
K tuples:none
Defined Rule Symbols:   minsort_1, appmin_3, notEmpty_1, remove_2, !EQ_2, <_2, remove[Ite]_3, appmin[Ite][True][Ite]_4

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_2, c14_2, c17_2


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

(61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   appmin[Ite][True][Ite](True, z0, Cons(z1, z2), z3) -> appmin(z1, z2, z3)
   appmin[Ite][True][Ite](False, z0, Cons(z1, z2), z3) -> appmin(z0, z2, z3)
   minsort(Cons(z0, z1)) -> appmin(z0, z1, Cons(z0, z1))
   minsort(Nil) -> Nil
   appmin(z0, Cons(z1, z2), z3) -> appmin[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2), z3)
   appmin(z0, Nil, z1) -> Cons(z0, minsort(remove(z0, z1)))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_2, c14_2, c17_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APPMIN(z0, Cons(z1, z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2), z3), <'(z1, z0)) by 
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3), <'(0, S(z0)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3), <'(z0, 0))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3), <'(0, S(z0)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3), <'(z0, 0))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3), <'(0, S(z0)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3), <'(z0, 0))
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c14_2, c17_2, c13_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, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c14_2, c17_2, c13_2, c13_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APPMIN(z0, Nil, z1) -> c14(MINSORT(remove(z0, z1)), REMOVE(z0, z1)) by 
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE_2, APPMIN_3

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c17_2, c13_2, c13_1, c14_2


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REMOVE(z0, Cons(z1, z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), z0, Cons(z1, z2)), !EQ'(z0, z1)) by 
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)), !EQ'(0, S(z0)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)), !EQ'(S(z0), 0))
   REMOVE(0, Cons(0, x2)) -> c17(REMOVE[ITE](True, 0, Cons(0, x2)), !EQ'(0, 0))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)), !EQ'(0, S(z0)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)), !EQ'(S(z0), 0))
   REMOVE(0, Cons(0, x2)) -> c17(REMOVE[ITE](True, 0, Cons(0, x2)), !EQ'(0, 0))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)), !EQ'(0, S(z0)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)), !EQ'(S(z0), 0))
   REMOVE(0, Cons(0, x2)) -> c17(REMOVE[ITE](True, 0, Cons(0, x2)), !EQ'(0, 0))
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_2, c13_1, c14_2, c17_2


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_2, c13_1, c14_2, c17_2, c17_1, c17


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APPMIN(S(z1), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(z1), Cons(S(z0), x2), x3), <'(S(z0), S(z1))) by 
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
K tuples:none
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c14_2, c17_2, c17_1, c17, c13_2


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

(75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
We considered the (Usable) Rules:
   <(z0, 0) -> False
   <(0, S(z0)) -> True
   <(S(z0), S(z1)) -> <(z0, z1)
And the Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(!EQ(x_1, x_2)) = x_1
   POL(!EQ'(x_1, x_2)) = 0
   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   POL(<'(x_1, x_2)) = 0
   POL(APPMIN(x_1, x_2, x_3)) = [1] + x_3
   POL(APPMIN[ITE][TRUE][ITE](x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(Cons(x_1, x_2)) = 0
   POL(False) = [1]
   POL(MINSORT(x_1)) = [1]
   POL(Nil) = [1]
   POL(REMOVE(x_1, x_2)) = 0
   POL(REMOVE[ITE](x_1, x_2, x_3)) = 0
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c17) = 0
   POL(c17(x_1)) = x_1
   POL(c17(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(remove(x_1, x_2)) = [1] + x_1 + x_2
   POL(remove[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2)))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c14_2, c17_2, c17_1, c17, c13_2


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APPMIN(z0, Nil, Cons(z1, z2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))), REMOVE(z0, Cons(z1, z2))) by 
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1)))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_2, c17_1, c17, c13_2, c14_2


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REMOVE(S(z0), Cons(S(z1), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(z0), Cons(S(z1), x2)), !EQ'(S(z0), S(z1))) by 
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(REMOVE[ITE](True, S(0), Cons(S(0), x2)), !EQ'(S(0), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(REMOVE[ITE](True, S(0), Cons(S(0), x2)), !EQ'(S(0), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(REMOVE[ITE](True, S(0), Cons(S(0), x2)), !EQ'(S(0), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2


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

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace APPMIN(0, Nil, Cons(S(z0), x2)) -> c14(MINSORT(remove[Ite](False, 0, Cons(S(z0), x2))), REMOVE(0, Cons(S(z0), x2))) by APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace APPMIN(S(z0), Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](False, S(z0), Cons(0, x2))), REMOVE(S(z0), Cons(0, x2))) by APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2


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

(87) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace APPMIN(0, Nil, Cons(0, x2)) -> c14(MINSORT(remove[Ite](True, 0, Cons(0, x2))), REMOVE(0, Cons(0, x2))) by APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2


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

(89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
We considered the (Usable) Rules:
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
And the Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(!EQ(x_1, x_2)) = x_1 + x_2
   POL(!EQ'(x_1, x_2)) = 0
   POL(0) = 0
   POL(<(x_1, x_2)) = [1] + x_1 + x_2
   POL(<'(x_1, x_2)) = 0
   POL(APPMIN(x_1, x_2, x_3)) = x_3
   POL(APPMIN[ITE][TRUE][ITE](x_1, x_2, x_3, x_4)) = x_4
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(MINSORT(x_1)) = x_1
   POL(Nil) = [1]
   POL(REMOVE(x_1, x_2)) = 0
   POL(REMOVE[ITE](x_1, x_2, x_3)) = 0
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c17) = 0
   POL(c17(x_1)) = x_1
   POL(c17(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(remove(x_1, x_2)) = x_2
   POL(remove[Ite](x_1, x_2, x_3)) = x_3

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   REMOVE[ITE](False, z0, Cons(z1, z2)) -> c7(REMOVE(z0, z2))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, REMOVE[ITE]_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2

Compound Symbols:   c_1, c4_1, c7_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2


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

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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   !EQ'(S(z0), S(z1)) -> c(!EQ'(z0, z1))
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   !EQ'_2, <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3

Compound Symbols:   c_1, c4_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1


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

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

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)), !EQ'(S(0), S(S(z0))))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)), !EQ'(S(S(z0)), S(0)))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17(!EQ'(S(0), S(0)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c4_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1


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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c4_1, c9_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN[ITE][TRUE][ITE](True, z0, Cons(z1, z2), z3) -> c9(APPMIN(z1, z2, z3)) by 
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c4_1, c10_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1


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

(99) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN[ITE][TRUE][ITE](False, z0, Cons(z1, z2), z3) -> c10(APPMIN(z0, z2, z3)) by 
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   <'_2, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4

Compound Symbols:   c4_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1, c10_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN(S(z0), Cons(0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, x2), x3)) by 
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
K tuples:
   APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   <'_2, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4

Compound Symbols:   c4_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1, c10_1


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

(103) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN(S(x0), Cons(S(x1), x2), x3) -> c13(<'(S(x1), S(x0))) by 
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   APPMIN(S(0), Cons(S(z1), z2), x2) -> c13(<'(S(z1), S(0)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   <'(S(z0), S(z1)) -> c4(<'(z0, z1))
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   APPMIN(S(0), Cons(S(z1), z2), x2) -> c13(<'(S(z1), S(0)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   APPMIN(S(0), Cons(S(z1), z2), x2) -> c13(<'(S(z1), S(0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   <'_2, MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4

Compound Symbols:   c4_1, c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1, c10_1


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace <'(S(z0), S(z1)) -> c4(<'(z0, z1)) by 
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   APPMIN(S(0), Cons(S(z1), z2), x2) -> c13(<'(S(z1), S(0)))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   APPMIN(S(0), Cons(S(z1), z2), x2) -> c13(<'(S(z1), S(0)))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2

Compound Symbols:   c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1, c10_1, c4_1


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

(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   APPMIN(S(0), Cons(S(z1), z2), x2) -> c13(<'(S(z1), S(0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3), <'(S(0), S(S(z0))))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3), <'(S(z0), S(0)))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2

Compound Symbols:   c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1, c10_1, c4_1


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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2)))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2

Compound Symbols:   c11_1, c13_1, c17_1, c17, c13_2, c14_2, c17_2, c7_1, c_1, c9_1, c10_1, c4_1


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

(111) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN(S(z0), Nil, Cons(S(z1), x2)) -> c14(MINSORT(remove[Ite](!EQ(z0, z1), S(z0), Cons(S(z1), x2))), REMOVE(S(z0), Cons(S(z1), x2))) by 
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2

Compound Symbols:   c11_1, c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1


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

(113) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN(S(z0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(z0), z1))), REMOVE(S(z0), Cons(0, z1))) by 
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2

Compound Symbols:   c11_1, c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1


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

(115) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2)) by 
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
S tuples:
   MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1)))
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   MINSORT_1, APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2

Compound Symbols:   c11_1, c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINSORT(Cons(z0, z1)) -> c11(APPMIN(z0, z1, Cons(z0, z1))) by 
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1)))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1, c11_1


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

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE(S(x0), Cons(S(x1), x2)) -> c17(!EQ'(S(x0), S(x1))) by 
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1, c11_1


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

(121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE[ITE](False, 0, Cons(S(x0), x1)) -> c7(REMOVE(0, x1)) by 
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2)))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1, c11_1


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

(123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE(0, Cons(S(z0), x2)) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), x2))) by 
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1, c11_1


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

(125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE[ITE](False, S(x0), Cons(0, x1)) -> c7(REMOVE(S(x0), x1)) by 
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2)))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c9_1, c10_1, c4_1, c11_1


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

(127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE(S(z0), Cons(0, x2)) -> c17(REMOVE[ITE](False, S(z0), Cons(0, x2))) by 
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c17_1, c9_1, c10_1, c4_1, c11_1


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

(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE[ITE](False, S(S(x0)), Cons(S(S(x1)), x2)) -> c7(REMOVE(S(S(x0)), x2)) by 
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c17_1, c9_1, c10_1, c4_1, c11_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE[ITE](False, S(0), Cons(S(S(x0)), x1)) -> c7(REMOVE(S(0), x1)) by 
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, REMOVE[ITE]_3, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1

Compound Symbols:   c13_1, c17, c13_2, c17_2, c14_2, c7_1, c_1, c17_1, c9_1, c10_1, c4_1, c11_1


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

(133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE[ITE](False, S(S(x0)), Cons(S(0), x1)) -> c7(REMOVE(S(S(x0)), x1)) by 
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, !EQ'_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3

Compound Symbols:   c13_1, c17, c13_2, c17_2, c14_2, c_1, c17_1, c9_1, c10_1, c4_1, c11_1, c7_1


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

(135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace !EQ'(S(S(y0)), S(S(y1))) -> c(!EQ'(S(y0), S(y1))) by 
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c17_2, c14_2, c17_1, c9_1, c10_1, c4_1, c11_1, c7_1, c_1


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

(137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE(S(S(z0)), Cons(S(S(z1)), x2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), x2)), !EQ'(S(S(z0)), S(S(z1)))) by 
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c14_2, c17_1, c9_1, c10_1, c4_1, c11_1, c7_1, c_1, c17_2


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

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE(S(0), Cons(S(S(z0)), x2)) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), x2))) by 
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c14_2, c17_1, c9_1, c10_1, c4_1, c11_1, c7_1, c_1, c17_2


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

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REMOVE(S(S(z0)), Cons(S(0), x2)) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), x2))) by 
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c14_2, c9_1, c10_1, c4_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c9(APPMIN(S(S(x1)), x2, x3)) by 
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c14_2, c9_1, c10_1, c4_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(S(0), x1), x2) -> c9(APPMIN(S(0), x1, x2)) by 
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c14_2, c10_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN[ITE][TRUE][ITE](False, 0, Cons(x0, x1), x2) -> c10(APPMIN(0, x1, x2)) by 
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
S tuples:
   APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3))
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   APPMIN_3, REMOVE_2, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c13_1, c17, c13_2, c14_2, c10_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(0, Cons(z0, x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, x2), x3)) by 
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c17, c13_2, c14_2, c10_1, c13_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN[ITE][TRUE][ITE](False, S(S(x0)), Cons(S(S(x1)), x2), x3) -> c10(APPMIN(S(S(x0)), x2, x3)) by 
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c17, c13_2, c14_2, c10_1, c13_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REMOVE(S(S(S(y0))), Cons(S(S(S(y1))), z2)) -> c17(REMOVE[ITE](!EQ(S(y0), S(y1)), S(S(S(y0))), Cons(S(S(S(y1))), z2)), !EQ'(S(S(S(y0))), S(S(S(y1))))) by REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2))
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, <'_2, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c17, c13_2, c14_2, c10_1, c13_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2


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

(155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(x0), x1), x2) -> c10(APPMIN(S(0), x1, x2)) by 
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c17, c13_2, c14_2, c13_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1


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

(157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(S(z0), Cons(S(z1), z2), Cons(S(z0), Cons(S(z1), z2))) -> c13(<'(S(z1), S(z0))) by 
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1))))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c17, c13_2, c14_2, c13_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1


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

(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(S(S(x1)), Cons(S(z1), z2), x3) -> c13(<'(S(z1), S(S(x1)))) by 
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1)))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, <'_2, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2

Compound Symbols:   c17, c13_2, c14_2, c13_1, c4_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1


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

(161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace <'(S(S(y0)), S(S(y1))) -> c4(<'(S(y0), S(y1))) by 
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1))))
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2, <'_2

Compound Symbols:   c17, c13_2, c14_2, c13_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1, c4_1


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

(163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(S(S(z1)), Cons(S(S(z0)), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](<(z0, z1), S(S(z1)), Cons(S(S(z0)), x2), x3), <'(S(S(z0)), S(S(z1)))) by 
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2, <'_2

Compound Symbols:   c17, c14_2, c13_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1, c4_1, c13_2


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(S(S(z0)), Cons(S(0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), x2), x3)) by 
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2, <'_2

Compound Symbols:   c17, c14_2, c13_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1, c4_1, c13_2


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(S(0), Cons(S(z0), x2), x3) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), x2), x3)) by 
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1))))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2, <'_2

Compound Symbols:   c17, c14_2, c13_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1, c4_1, c13_2


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN[ITE][TRUE][ITE](True, S(x0), Cons(0, x1), Cons(S(x0), Cons(0, x1))) -> c9(APPMIN(0, x1, Cons(S(x0), Cons(0, x1)))) by 
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))) -> c9(APPMIN(0, Nil, Cons(S(z0), Cons(0, Nil))))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y0, Cons(y1, y2))), Cons(S(z0), Cons(0, Cons(y0, Cons(y1, y2))))) -> c9(APPMIN(0, Cons(y0, Cons(y1, y2)), Cons(S(z0), Cons(0, Cons(y0, Cons(y1, y2))))))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y0, Nil)), Cons(S(z0), Cons(0, Cons(y0, Nil)))) -> c9(APPMIN(0, Cons(y0, Nil), Cons(S(z0), Cons(0, Cons(y0, Nil)))))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))) -> c9(APPMIN(0, Nil, Cons(S(z0), Cons(0, Nil))))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y0, Cons(y1, y2))), Cons(S(z0), Cons(0, Cons(y0, Cons(y1, y2))))) -> c9(APPMIN(0, Cons(y0, Cons(y1, y2)), Cons(S(z0), Cons(0, Cons(y0, Cons(y1, y2))))))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y0, Nil)), Cons(S(z0), Cons(0, Cons(y0, Nil)))) -> c9(APPMIN(0, Cons(y0, Nil), Cons(S(z0), Cons(0, Cons(y0, Nil)))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))))
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2, <'_2

Compound Symbols:   c17, c14_2, c13_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1, c4_1, c13_2


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APPMIN(S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, z1), Cons(S(z0), Cons(0, z1)))) by 
   APPMIN(S(S(y0)), Cons(0, z1), Cons(S(S(y0)), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(y0)), Cons(0, z1), Cons(S(S(y0)), Cons(0, z1))))
   APPMIN(S(0), Cons(0, z1), Cons(S(0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), Cons(S(0), Cons(0, z1))))
   APPMIN(S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))))
   APPMIN(S(z0), Cons(0, Cons(y1, Cons(y2, y3))), Cons(S(z0), Cons(0, Cons(y1, Cons(y2, y3))))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y1, Cons(y2, y3))), Cons(S(z0), Cons(0, Cons(y1, Cons(y2, y3))))))
   APPMIN(S(z0), Cons(0, Cons(y1, Nil)), Cons(S(z0), Cons(0, Cons(y1, Nil)))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y1, Nil)), Cons(S(z0), Cons(0, Cons(y1, Nil)))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   remove(z0, Cons(z1, z2)) -> remove[Ite](!EQ(z0, z1), z0, Cons(z1, z2))
   remove[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, remove(z0, z2))
   remove[Ite](True, z0, Cons(z1, z2)) -> z2
   !EQ(S(z0), S(z1)) -> !EQ(z0, z1)
   !EQ(0, S(z0)) -> False
   !EQ(S(z0), 0) -> False
   !EQ(0, 0) -> True
Tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(x0)), Cons(0, x1), x2) -> c9(APPMIN(0, x1, x2))
   APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, x0), x1) -> c9(APPMIN(0, x0, x1))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y0), y1))) -> c7(REMOVE(0, Cons(S(y0), y1)))
   REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y0))) -> c7(REMOVE(0, Cons(0, y0)))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))) -> c7(REMOVE(S(z0), Cons(0, y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(y0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))) -> c7(REMOVE(S(S(y0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y0)), y1))) -> c7(REMOVE(S(0), Cons(S(S(y0)), y1)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y0))) -> c7(REMOVE(S(0), Cons(S(0), y0)))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y0)), y1)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y0)))) -> c7(REMOVE(S(0), Cons(0, Cons(S(0), y0))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c7(REMOVE(S(S(z0)), Cons(S(S(y1)), y2)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c7(REMOVE(S(S(z0)), Cons(S(0), y1)))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(0, y1))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c7(REMOVE(S(S(z0)), Cons(0, Cons(S(0), y1))))
   !EQ'(S(S(S(y0))), S(S(S(y1)))) -> c(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(S(z1)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c9(APPMIN(S(S(z1)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c9(APPMIN(S(S(z1)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c9(APPMIN(S(S(z1)), Nil, Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c9(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y0)), z2) -> c9(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c9(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y0), y1)), z2) -> c9(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c9(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y0), y1)) -> c9(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y0)) -> c9(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y0, y1)), z2) -> c10(APPMIN(0, Cons(y0, y1), z2))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y0), y1)) -> c10(APPMIN(0, Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y0)) -> c10(APPMIN(0, Nil, Cons(0, y0)))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(S(y1)), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(S(y1)), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(S(z0)), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(0, y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(0, y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(y1), y2)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(y1), y2), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Cons(S(0), y1)), z3) -> c10(APPMIN(S(S(z0)), Cons(S(0), y1), z3))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y1), y2)) -> c10(APPMIN(S(S(z0)), Nil, Cons(S(y1), y2)))
   APPMIN[ITE][TRUE][ITE](False, S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y1)) -> c10(APPMIN(S(S(z0)), Nil, Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c10(APPMIN(S(0), Cons(0, y1), Cons(S(y2), Cons(0, y3))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y0)), z2) -> c10(APPMIN(S(0), Cons(0, y0), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c10(APPMIN(S(0), Cons(S(y1), y2), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y0), y1)), z2) -> c10(APPMIN(S(0), Cons(S(y0), y1), z2))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c10(APPMIN(S(0), Nil, Cons(S(y1), Nil)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y0), y1)) -> c10(APPMIN(S(0), Nil, Cons(S(y0), y1)))
   APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y0)) -> c10(APPMIN(S(0), Nil, Cons(0, y0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
   <'(S(S(S(y0))), S(S(S(y1)))) -> c4(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))) -> c9(APPMIN(0, Nil, Cons(S(z0), Cons(0, Nil))))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y0, Cons(y1, y2))), Cons(S(z0), Cons(0, Cons(y0, Cons(y1, y2))))) -> c9(APPMIN(0, Cons(y0, Cons(y1, y2)), Cons(S(z0), Cons(0, Cons(y0, Cons(y1, y2))))))
   APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y0, Nil)), Cons(S(z0), Cons(0, Cons(y0, Nil)))) -> c9(APPMIN(0, Cons(y0, Nil), Cons(S(z0), Cons(0, Cons(y0, Nil)))))
   APPMIN(S(S(y0)), Cons(0, z1), Cons(S(S(y0)), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(y0)), Cons(0, z1), Cons(S(S(y0)), Cons(0, z1))))
   APPMIN(S(0), Cons(0, z1), Cons(S(0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), Cons(S(0), Cons(0, z1))))
   APPMIN(S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))))
   APPMIN(S(z0), Cons(0, Cons(y1, Cons(y2, y3))), Cons(S(z0), Cons(0, Cons(y1, Cons(y2, y3))))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y1, Cons(y2, y3))), Cons(S(z0), Cons(0, Cons(y1, Cons(y2, y3))))))
   APPMIN(S(z0), Cons(0, Cons(y1, Nil)), Cons(S(z0), Cons(0, Cons(y1, Nil)))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y1, Nil)), Cons(S(z0), Cons(0, Cons(y1, Nil)))))
S tuples:
   REMOVE(0, Cons(0, x2)) -> c17
   APPMIN(0, Nil, Cons(S(z0), z1)) -> c14(MINSORT(Cons(S(z0), remove(0, z1))), REMOVE(0, Cons(S(z0), z1)))
   REMOVE(S(0), Cons(S(0), x2)) -> c17
   APPMIN(S(S(x1)), Cons(0, z1), x3) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(x1)), Cons(0, z1), x3))
   APPMIN(S(0), Cons(0, z1), x2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), x2))
   APPMIN(S(z0), Nil, Cons(S(z0), Nil)) -> c14(MINSORT(remove[Ite](!EQ(z0, z0), S(z0), Cons(S(z0), Nil))), REMOVE(S(z0), Cons(S(z0), Nil)))
   APPMIN(S(S(x1)), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(S(x1), z1), S(S(x1)), Cons(S(z1), z2))), REMOVE(S(S(x1)), Cons(S(z1), z2)))
   APPMIN(S(0), Nil, Cons(S(z1), z2)) -> c14(MINSORT(remove[Ite](!EQ(0, z1), S(0), Cons(S(z1), z2))), REMOVE(S(0), Cons(S(z1), z2)))
   APPMIN(S(S(x1)), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(S(x1)), z1))), REMOVE(S(S(x1)), Cons(0, z1)))
   APPMIN(S(0), Nil, Cons(0, z1)) -> c14(MINSORT(Cons(0, remove(S(0), z1))), REMOVE(S(0), Cons(0, z1)))
   MINSORT(Cons(0, Cons(y0, y1))) -> c11(APPMIN(0, Cons(y0, y1), Cons(0, Cons(y0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(S(S(y1)), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(S(y1)), y2), Cons(S(S(y0)), Cons(S(S(y1)), y2))))
   MINSORT(Cons(0, Nil)) -> c11(APPMIN(0, Nil, Cons(0, Nil)))
   MINSORT(Cons(S(y0), Cons(0, y1))) -> c11(APPMIN(S(y0), Cons(0, y1), Cons(S(y0), Cons(0, y1))))
   MINSORT(Cons(S(S(y0)), Cons(0, y1))) -> c11(APPMIN(S(S(y0)), Cons(0, y1), Cons(S(S(y0)), Cons(0, y1))))
   MINSORT(Cons(S(0), Cons(0, y0))) -> c11(APPMIN(S(0), Cons(0, y0), Cons(S(0), Cons(0, y0))))
   MINSORT(Cons(S(y0), Cons(S(y1), y2))) -> c11(APPMIN(S(y0), Cons(S(y1), y2), Cons(S(y0), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(y1), y2))) -> c11(APPMIN(S(S(y0)), Cons(S(y1), y2), Cons(S(S(y0)), Cons(S(y1), y2))))
   MINSORT(Cons(S(S(y0)), Cons(S(0), y1))) -> c11(APPMIN(S(S(y0)), Cons(S(0), y1), Cons(S(S(y0)), Cons(S(0), y1))))
   MINSORT(Cons(S(0), Cons(S(y0), y1))) -> c11(APPMIN(S(0), Cons(S(y0), y1), Cons(S(0), Cons(S(y0), y1))))
   MINSORT(Cons(S(y0), Nil)) -> c11(APPMIN(S(y0), Nil, Cons(S(y0), Nil)))
   MINSORT(Cons(S(S(y0)), Nil)) -> c11(APPMIN(S(S(y0)), Nil, Cons(S(S(y0)), Nil)))
   MINSORT(Cons(S(0), Nil)) -> c11(APPMIN(S(0), Nil, Cons(S(0), Nil)))
   REMOVE(S(S(y0)), Cons(S(S(y1)), z2)) -> c17(!EQ'(S(S(y0)), S(S(y1))))
   REMOVE(0, Cons(S(z0), Cons(S(y1), y2))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(S(y1), y2))))
   REMOVE(0, Cons(S(z0), Cons(0, y1))) -> c17(REMOVE[ITE](False, 0, Cons(S(z0), Cons(0, y1))))
   REMOVE(S(z0), Cons(0, Cons(0, y1))) -> c17(REMOVE[ITE](False, S(z0), Cons(0, Cons(0, y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(0, Cons(S(S(y0)), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(S(y0)), y1))))
   REMOVE(S(S(y0)), Cons(0, Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(y0)), Cons(0, Cons(S(0), y1))))
   REMOVE(S(0), Cons(0, Cons(S(0), y0))) -> c17(REMOVE[ITE](False, S(0), Cons(0, Cons(S(0), y0))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(0, y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(S(y3)), y4)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(z0)), Cons(S(S(z1)), Cons(0, Cons(S(0), y3)))), !EQ'(S(S(z0)), S(S(z1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(S(y1)), y2))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(S(0), y1))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(0, y1)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(0), Cons(S(S(z0)), Cons(0, Cons(S(0), y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(S(y1)), y2))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(S(0), y1))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(S(0), y1))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(0, y1)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(S(y1)), y2)))))
   REMOVE(S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))) -> c17(REMOVE[ITE](False, S(S(z0)), Cons(S(0), Cons(0, Cons(S(0), y1)))))
   APPMIN(0, Cons(z0, Cons(y1, y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Cons(y1, y2)), z2))
   APPMIN(0, Cons(z0, Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(S(y1), y2)))
   APPMIN(0, Cons(z0, Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, 0, Cons(z0, Nil), Cons(0, y1)))
   REMOVE(S(S(S(z0))), Cons(S(S(S(z1))), z2)) -> c17(REMOVE[ITE](!EQ(z0, z1), S(S(S(z0))), Cons(S(S(S(z1))), z2)), !EQ'(S(S(S(z0))), S(S(S(z1)))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(S(y3)), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), Cons(S(y4), Cons(0, y5))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(0, y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), Cons(S(y5), Cons(S(y6), y7))), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(y3), y4)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Cons(S(0), y3)), z3), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), Nil)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(S(y3), y4)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)) -> c13(APPMIN[ITE][TRUE][ITE](<(z1, z0), S(S(z0)), Cons(S(S(z1)), Nil), Cons(0, y3)), <'(S(S(z1)), S(S(z0))))
   APPMIN(S(S(S(y1))), Cons(S(S(S(y0))), z2), z3) -> c13(APPMIN[ITE][TRUE][ITE](<(S(y0), S(y1)), S(S(S(y1))), Cons(S(S(S(y0))), z2), z3), <'(S(S(S(y0))), S(S(S(y1)))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(0, y1)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Cons(S(y1), y2)), z2))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(S(y1), y2)))
   APPMIN(S(S(z0)), Cons(S(0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(z0)), Cons(S(0), Nil), Cons(0, y1)))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), Cons(S(y2), Cons(0, y3))))
   APPMIN(S(0), Cons(S(z0), Cons(0, y1)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(0, y1)), z2))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), Cons(S(y3), Cons(S(y4), y5))))
   APPMIN(S(0), Cons(S(z0), Cons(S(y1), y2)), z2) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Cons(S(y1), y2)), z2))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), Nil)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(S(y1), y2)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(S(y1), y2)))
   APPMIN(S(0), Cons(S(z0), Nil), Cons(0, y1)) -> c13(APPMIN[ITE][TRUE][ITE](False, S(0), Cons(S(z0), Nil), Cons(0, y1)))
   APPMIN(S(S(y0)), Cons(0, z1), Cons(S(S(y0)), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(S(y0)), Cons(0, z1), Cons(S(S(y0)), Cons(0, z1))))
   APPMIN(S(0), Cons(0, z1), Cons(S(0), Cons(0, z1))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(0), Cons(0, z1), Cons(S(0), Cons(0, z1))))
   APPMIN(S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Nil), Cons(S(z0), Cons(0, Nil))))
   APPMIN(S(z0), Cons(0, Cons(y1, Cons(y2, y3))), Cons(S(z0), Cons(0, Cons(y1, Cons(y2, y3))))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y1, Cons(y2, y3))), Cons(S(z0), Cons(0, Cons(y1, Cons(y2, y3))))))
   APPMIN(S(z0), Cons(0, Cons(y1, Nil)), Cons(S(z0), Cons(0, Cons(y1, Nil)))) -> c13(APPMIN[ITE][TRUE][ITE](True, S(z0), Cons(0, Cons(y1, Nil)), Cons(S(z0), Cons(0, Cons(y1, Nil)))))
K tuples:
   APPMIN(0, Nil, Cons(0, z0)) -> c14(MINSORT(z0), REMOVE(0, Cons(0, z0)))
   APPMIN(S(S(y1)), Cons(S(S(y0)), z2), Cons(S(S(y1)), Cons(S(S(y0)), z2))) -> c13(<'(S(S(y0)), S(S(y1))))
   APPMIN(S(S(z0)), Cons(S(S(y0)), z2), z3) -> c13(<'(S(S(y0)), S(S(z0))))
Defined Rule Symbols:   <_2, remove_2, remove[Ite]_3, !EQ_2

Defined Pair Symbols:   REMOVE_2, APPMIN_3, APPMIN[ITE][TRUE][ITE]_4, MINSORT_1, REMOVE[ITE]_3, !EQ'_2, <'_2

Compound Symbols:   c17, c14_2, c13_1, c9_1, c11_1, c17_1, c7_1, c_1, c17_2, c10_1, c4_1, c13_2