KILLED
proof of input_5bzcYnwoGk.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 713 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 334 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 364 ms]
            (24) typed CpxTrs
(25) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(26) CpxWeightedTrs
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxTypedWeightedTrs
    (29) CompletionProof [UPPER BOUND(ID), 0 ms]
    (30) CpxTypedWeightedCompleteTrs
        (31) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms]
        (34) CpxRNTS
        (35) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (44) CdtProblem
    (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 4 ms]
    (48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
        (51) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (52) CpxTRS
        (53) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CpxWeightedTrs
            (55) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (56) CpxTypedWeightedTrs
            (57) CompletionProof [UPPER BOUND(ID), 0 ms]
            (58) CpxTypedWeightedCompleteTrs
                (59) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (60) CpxTypedWeightedCompleteTrs
                (61) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (62) CpxRNTS
                (63) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (64) CpxRNTS
            (65) CompletionProof [UPPER BOUND(ID), 0 ms]
            (66) CpxTypedWeightedCompleteTrs
                (67) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (68) CpxRNTS
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
        (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CdtProblem
        (73) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CdtProblem
        (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem
        (81) CdtNarrowingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (96) CdtProblem
        (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
        (98) CdtProblem
        (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem
        (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem
        (111) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem
        (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (116) CdtProblem
        (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (118) CdtProblem
        (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (120) CdtProblem
        (121) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CdtProblem
        (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (126) CdtProblem
        (127) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
        (130) CdtProblem
        (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (134) CdtProblem
        (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (136) CdtProblem
        (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (140) CdtProblem
        (141) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (142) CdtProblem
        (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (144) CdtProblem
        (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms]
        (148) CdtProblem
        (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (150) CdtProblem
        (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (152) CdtProblem
        (153) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
        (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (156) CdtProblem
        (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (158) CdtProblem
        (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (160) CdtProblem
        (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (162) CdtProblem
        (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (164) CdtProblem
        (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (166) CdtProblem
        (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (168) CdtProblem
        (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (170) CdtProblem
        (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (172) CdtProblem
        (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (174) CdtProblem
        (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (176) CdtProblem
        (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (178) CdtProblem
        (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CdtProblem
        (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CdtProblem
        (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (184) CdtProblem
        (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (186) CdtProblem
        (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (188) CdtProblem
        (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
        (190) CdtProblem
        (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (192) CdtProblem
        (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (194) CdtProblem
        (195) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (196) CdtProblem


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

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


The TRS R consists of the following rules:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

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

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


The TRS R consists of the following rules:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0'
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0') -> 0'
   p(s(x)) -> x
   ack(0', x) -> plus(x, s(0'))
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0', y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0')) -> s(x)
   plus(x, 0') -> x
   isZero(0') -> true
   isZero(s(x)) -> false

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

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

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


The TRS R consists of the following rules:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18
S tuples:
   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18
K tuples:none
Defined Rule Symbols:   double_1, permute_3, p_1, ack_2, plus_2, isZero_1

Defined Pair Symbols:   DOUBLE_1, PERMUTE_3, P_1, ACK_2, PLUS_2, ISZERO_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5, c6_1, c7, c8, c9_1, c10_1, c11_2, c12, c13_1, c14_1, c15, c16, c17, c18


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0') -> c7
   P(s(z0)) -> c8
   ACK(0', z0) -> c9(PLUS(z0, s(0')))
   ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0', z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0')) -> c15
   PLUS(z0, 0') -> c16
   ISZERO(0') -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0'
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0') -> 0'
   p(s(z0)) -> z0
   ack(0', z0) -> plus(z0, s(0'))
   ack(s(z0), 0') -> ack(z0, s(0'))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0', z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0')) -> s(z0)
   plus(z0, 0') -> z0
   isZero(0') -> true
   isZero(s(z0)) -> false

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

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

(12)
Obligation:
Innermost TRS:
Rules:
DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
PERMUTE(true, z0, b) -> c5
PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
P(0') -> c7
P(s(z0)) -> c8
ACK(0', z0) -> c9(PLUS(z0, s(0')))
ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
PLUS(0', z0) -> c12
PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
PLUS(z0, s(0')) -> c15
PLUS(z0, 0') -> c16
ISZERO(0') -> c17
ISZERO(s(z0)) -> c18
double(z0) -> permute(z0, z0, a)
permute(z0, z1, a) -> permute(isZero(z0), z0, b)
permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
permute(true, z0, b) -> 0'
permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
p(0') -> 0'
p(s(z0)) -> z0
ack(0', z0) -> plus(z0, s(0'))
ack(s(z0), 0') -> ack(z0, s(0'))
ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
plus(z0, s(0')) -> s(z0)
plus(z0, 0') -> z0
isZero(0') -> true
isZero(s(z0)) -> false

Types:
DOUBLE :: false:true:0':s -> c1
c1 :: c2:c3:c4:c5:c6 -> c1
PERMUTE :: false:true:0':s -> false:true:0':s -> a:b:c -> c2:c3:c4:c5:c6
a :: a:b:c
c2 :: c2:c3:c4:c5:c6 -> c17:c18 -> c2:c3:c4:c5:c6
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
ISZERO :: false:true:0':s -> c17:c18
false :: false:true:0':s
c3 :: c2:c3:c4:c5:c6 -> c9:c10:c11 -> c2:c3:c4:c5:c6
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
ACK :: false:true:0':s -> false:true:0':s -> c9:c10:c11
c4 :: c2:c3:c4:c5:c6 -> c7:c8 -> c2:c3:c4:c5:c6
P :: false:true:0':s -> c7:c8
true :: false:true:0':s
c5 :: c2:c3:c4:c5:c6
c6 :: c2:c3:c4:c5:c6 -> c2:c3:c4:c5:c6
0' :: false:true:0':s
c7 :: c7:c8
s :: false:true:0':s -> false:true:0':s
c8 :: c7:c8
c9 :: c12:c13:c14:c15:c16 -> c9:c10:c11
PLUS :: false:true:0':s -> false:true:0':s -> c12:c13:c14:c15:c16
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14:c15:c16
c13 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c14 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c15 :: c12:c13:c14:c15:c16
c16 :: c12:c13:c14:c15:c16
c17 :: c17:c18
c18 :: c17:c18
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_c11_19 :: c1
hole_false:true:0':s2_19 :: false:true:0':s
hole_c2:c3:c4:c5:c63_19 :: c2:c3:c4:c5:c6
hole_a:b:c4_19 :: a:b:c
hole_c17:c185_19 :: c17:c18
hole_c9:c10:c116_19 :: c9:c10:c11
hole_c7:c87_19 :: c7:c8
hole_c12:c13:c14:c15:c168_19 :: c12:c13:c14:c15:c16
gen_false:true:0':s9_19 :: Nat -> false:true:0':s
gen_c2:c3:c4:c5:c610_19 :: Nat -> c2:c3:c4:c5:c6
gen_c9:c10:c1111_19 :: Nat -> c9:c10:c11
gen_c12:c13:c14:c15:c1612_19 :: Nat -> c12:c13:c14:c15:c16

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PERMUTE, ack, ACK, PLUS, permute, plus

They will be analysed ascendingly in the following order:
ack < PERMUTE
ACK < PERMUTE
ack < ACK
ack < permute
plus < ack
PLUS < ACK

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

(14)
Obligation:
Innermost TRS:
Rules:
DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
PERMUTE(true, z0, b) -> c5
PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
P(0') -> c7
P(s(z0)) -> c8
ACK(0', z0) -> c9(PLUS(z0, s(0')))
ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
PLUS(0', z0) -> c12
PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
PLUS(z0, s(0')) -> c15
PLUS(z0, 0') -> c16
ISZERO(0') -> c17
ISZERO(s(z0)) -> c18
double(z0) -> permute(z0, z0, a)
permute(z0, z1, a) -> permute(isZero(z0), z0, b)
permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
permute(true, z0, b) -> 0'
permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
p(0') -> 0'
p(s(z0)) -> z0
ack(0', z0) -> plus(z0, s(0'))
ack(s(z0), 0') -> ack(z0, s(0'))
ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
plus(z0, s(0')) -> s(z0)
plus(z0, 0') -> z0
isZero(0') -> true
isZero(s(z0)) -> false

Types:
DOUBLE :: false:true:0':s -> c1
c1 :: c2:c3:c4:c5:c6 -> c1
PERMUTE :: false:true:0':s -> false:true:0':s -> a:b:c -> c2:c3:c4:c5:c6
a :: a:b:c
c2 :: c2:c3:c4:c5:c6 -> c17:c18 -> c2:c3:c4:c5:c6
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
ISZERO :: false:true:0':s -> c17:c18
false :: false:true:0':s
c3 :: c2:c3:c4:c5:c6 -> c9:c10:c11 -> c2:c3:c4:c5:c6
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
ACK :: false:true:0':s -> false:true:0':s -> c9:c10:c11
c4 :: c2:c3:c4:c5:c6 -> c7:c8 -> c2:c3:c4:c5:c6
P :: false:true:0':s -> c7:c8
true :: false:true:0':s
c5 :: c2:c3:c4:c5:c6
c6 :: c2:c3:c4:c5:c6 -> c2:c3:c4:c5:c6
0' :: false:true:0':s
c7 :: c7:c8
s :: false:true:0':s -> false:true:0':s
c8 :: c7:c8
c9 :: c12:c13:c14:c15:c16 -> c9:c10:c11
PLUS :: false:true:0':s -> false:true:0':s -> c12:c13:c14:c15:c16
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14:c15:c16
c13 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c14 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c15 :: c12:c13:c14:c15:c16
c16 :: c12:c13:c14:c15:c16
c17 :: c17:c18
c18 :: c17:c18
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_c11_19 :: c1
hole_false:true:0':s2_19 :: false:true:0':s
hole_c2:c3:c4:c5:c63_19 :: c2:c3:c4:c5:c6
hole_a:b:c4_19 :: a:b:c
hole_c17:c185_19 :: c17:c18
hole_c9:c10:c116_19 :: c9:c10:c11
hole_c7:c87_19 :: c7:c8
hole_c12:c13:c14:c15:c168_19 :: c12:c13:c14:c15:c16
gen_false:true:0':s9_19 :: Nat -> false:true:0':s
gen_c2:c3:c4:c5:c610_19 :: Nat -> c2:c3:c4:c5:c6
gen_c9:c10:c1111_19 :: Nat -> c9:c10:c11
gen_c12:c13:c14:c15:c1612_19 :: Nat -> c12:c13:c14:c15:c16


Generator Equations:
gen_false:true:0':s9_19(0) <=> true
gen_false:true:0':s9_19(+(x, 1)) <=> s(gen_false:true:0':s9_19(x))
gen_c2:c3:c4:c5:c610_19(0) <=> c5
gen_c2:c3:c4:c5:c610_19(+(x, 1)) <=> c2(gen_c2:c3:c4:c5:c610_19(x), c17)
gen_c9:c10:c1111_19(0) <=> c9(c12)
gen_c9:c10:c1111_19(+(x, 1)) <=> c10(gen_c9:c10:c1111_19(x))
gen_c12:c13:c14:c15:c1612_19(0) <=> c12
gen_c12:c13:c14:c15:c1612_19(+(x, 1)) <=> c13(gen_c12:c13:c14:c15:c1612_19(x))


The following defined symbols remain to be analysed:
PLUS, PERMUTE, ack, ACK, permute, plus

They will be analysed ascendingly in the following order:
ack < PERMUTE
ACK < PERMUTE
ack < ACK
ack < permute
plus < ack
PLUS < ACK

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n14_19)))) -> *13_19, rt in Omega(n14_19)

Induction Base:
PLUS(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, 0))))

Induction Step:
PLUS(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, +(n14_19, 1))))) ->_R^Omega(1)
c14(PLUS(s(gen_false:true:0':s9_19(a)), gen_false:true:0':s9_19(+(2, *(2, n14_19))))) ->_IH
c14(*13_19)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
PERMUTE(true, z0, b) -> c5
PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
P(0') -> c7
P(s(z0)) -> c8
ACK(0', z0) -> c9(PLUS(z0, s(0')))
ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
PLUS(0', z0) -> c12
PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
PLUS(z0, s(0')) -> c15
PLUS(z0, 0') -> c16
ISZERO(0') -> c17
ISZERO(s(z0)) -> c18
double(z0) -> permute(z0, z0, a)
permute(z0, z1, a) -> permute(isZero(z0), z0, b)
permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
permute(true, z0, b) -> 0'
permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
p(0') -> 0'
p(s(z0)) -> z0
ack(0', z0) -> plus(z0, s(0'))
ack(s(z0), 0') -> ack(z0, s(0'))
ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
plus(z0, s(0')) -> s(z0)
plus(z0, 0') -> z0
isZero(0') -> true
isZero(s(z0)) -> false

Types:
DOUBLE :: false:true:0':s -> c1
c1 :: c2:c3:c4:c5:c6 -> c1
PERMUTE :: false:true:0':s -> false:true:0':s -> a:b:c -> c2:c3:c4:c5:c6
a :: a:b:c
c2 :: c2:c3:c4:c5:c6 -> c17:c18 -> c2:c3:c4:c5:c6
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
ISZERO :: false:true:0':s -> c17:c18
false :: false:true:0':s
c3 :: c2:c3:c4:c5:c6 -> c9:c10:c11 -> c2:c3:c4:c5:c6
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
ACK :: false:true:0':s -> false:true:0':s -> c9:c10:c11
c4 :: c2:c3:c4:c5:c6 -> c7:c8 -> c2:c3:c4:c5:c6
P :: false:true:0':s -> c7:c8
true :: false:true:0':s
c5 :: c2:c3:c4:c5:c6
c6 :: c2:c3:c4:c5:c6 -> c2:c3:c4:c5:c6
0' :: false:true:0':s
c7 :: c7:c8
s :: false:true:0':s -> false:true:0':s
c8 :: c7:c8
c9 :: c12:c13:c14:c15:c16 -> c9:c10:c11
PLUS :: false:true:0':s -> false:true:0':s -> c12:c13:c14:c15:c16
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14:c15:c16
c13 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c14 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c15 :: c12:c13:c14:c15:c16
c16 :: c12:c13:c14:c15:c16
c17 :: c17:c18
c18 :: c17:c18
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_c11_19 :: c1
hole_false:true:0':s2_19 :: false:true:0':s
hole_c2:c3:c4:c5:c63_19 :: c2:c3:c4:c5:c6
hole_a:b:c4_19 :: a:b:c
hole_c17:c185_19 :: c17:c18
hole_c9:c10:c116_19 :: c9:c10:c11
hole_c7:c87_19 :: c7:c8
hole_c12:c13:c14:c15:c168_19 :: c12:c13:c14:c15:c16
gen_false:true:0':s9_19 :: Nat -> false:true:0':s
gen_c2:c3:c4:c5:c610_19 :: Nat -> c2:c3:c4:c5:c6
gen_c9:c10:c1111_19 :: Nat -> c9:c10:c11
gen_c12:c13:c14:c15:c1612_19 :: Nat -> c12:c13:c14:c15:c16


Generator Equations:
gen_false:true:0':s9_19(0) <=> true
gen_false:true:0':s9_19(+(x, 1)) <=> s(gen_false:true:0':s9_19(x))
gen_c2:c3:c4:c5:c610_19(0) <=> c5
gen_c2:c3:c4:c5:c610_19(+(x, 1)) <=> c2(gen_c2:c3:c4:c5:c610_19(x), c17)
gen_c9:c10:c1111_19(0) <=> c9(c12)
gen_c9:c10:c1111_19(+(x, 1)) <=> c10(gen_c9:c10:c1111_19(x))
gen_c12:c13:c14:c15:c1612_19(0) <=> c12
gen_c12:c13:c14:c15:c1612_19(+(x, 1)) <=> c13(gen_c12:c13:c14:c15:c1612_19(x))


The following defined symbols remain to be analysed:
PLUS, PERMUTE, ack, ACK, permute, plus

They will be analysed ascendingly in the following order:
ack < PERMUTE
ACK < PERMUTE
ack < ACK
ack < permute
plus < ack
PLUS < ACK

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

(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
PERMUTE(true, z0, b) -> c5
PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
P(0') -> c7
P(s(z0)) -> c8
ACK(0', z0) -> c9(PLUS(z0, s(0')))
ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
PLUS(0', z0) -> c12
PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
PLUS(z0, s(0')) -> c15
PLUS(z0, 0') -> c16
ISZERO(0') -> c17
ISZERO(s(z0)) -> c18
double(z0) -> permute(z0, z0, a)
permute(z0, z1, a) -> permute(isZero(z0), z0, b)
permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
permute(true, z0, b) -> 0'
permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
p(0') -> 0'
p(s(z0)) -> z0
ack(0', z0) -> plus(z0, s(0'))
ack(s(z0), 0') -> ack(z0, s(0'))
ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
plus(z0, s(0')) -> s(z0)
plus(z0, 0') -> z0
isZero(0') -> true
isZero(s(z0)) -> false

Types:
DOUBLE :: false:true:0':s -> c1
c1 :: c2:c3:c4:c5:c6 -> c1
PERMUTE :: false:true:0':s -> false:true:0':s -> a:b:c -> c2:c3:c4:c5:c6
a :: a:b:c
c2 :: c2:c3:c4:c5:c6 -> c17:c18 -> c2:c3:c4:c5:c6
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
ISZERO :: false:true:0':s -> c17:c18
false :: false:true:0':s
c3 :: c2:c3:c4:c5:c6 -> c9:c10:c11 -> c2:c3:c4:c5:c6
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
ACK :: false:true:0':s -> false:true:0':s -> c9:c10:c11
c4 :: c2:c3:c4:c5:c6 -> c7:c8 -> c2:c3:c4:c5:c6
P :: false:true:0':s -> c7:c8
true :: false:true:0':s
c5 :: c2:c3:c4:c5:c6
c6 :: c2:c3:c4:c5:c6 -> c2:c3:c4:c5:c6
0' :: false:true:0':s
c7 :: c7:c8
s :: false:true:0':s -> false:true:0':s
c8 :: c7:c8
c9 :: c12:c13:c14:c15:c16 -> c9:c10:c11
PLUS :: false:true:0':s -> false:true:0':s -> c12:c13:c14:c15:c16
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14:c15:c16
c13 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c14 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c15 :: c12:c13:c14:c15:c16
c16 :: c12:c13:c14:c15:c16
c17 :: c17:c18
c18 :: c17:c18
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_c11_19 :: c1
hole_false:true:0':s2_19 :: false:true:0':s
hole_c2:c3:c4:c5:c63_19 :: c2:c3:c4:c5:c6
hole_a:b:c4_19 :: a:b:c
hole_c17:c185_19 :: c17:c18
hole_c9:c10:c116_19 :: c9:c10:c11
hole_c7:c87_19 :: c7:c8
hole_c12:c13:c14:c15:c168_19 :: c12:c13:c14:c15:c16
gen_false:true:0':s9_19 :: Nat -> false:true:0':s
gen_c2:c3:c4:c5:c610_19 :: Nat -> c2:c3:c4:c5:c6
gen_c9:c10:c1111_19 :: Nat -> c9:c10:c11
gen_c12:c13:c14:c15:c1612_19 :: Nat -> c12:c13:c14:c15:c16


Lemmas:
PLUS(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n14_19)))) -> *13_19, rt in Omega(n14_19)


Generator Equations:
gen_false:true:0':s9_19(0) <=> true
gen_false:true:0':s9_19(+(x, 1)) <=> s(gen_false:true:0':s9_19(x))
gen_c2:c3:c4:c5:c610_19(0) <=> c5
gen_c2:c3:c4:c5:c610_19(+(x, 1)) <=> c2(gen_c2:c3:c4:c5:c610_19(x), c17)
gen_c9:c10:c1111_19(0) <=> c9(c12)
gen_c9:c10:c1111_19(+(x, 1)) <=> c10(gen_c9:c10:c1111_19(x))
gen_c12:c13:c14:c15:c1612_19(0) <=> c12
gen_c12:c13:c14:c15:c1612_19(+(x, 1)) <=> c13(gen_c12:c13:c14:c15:c1612_19(x))


The following defined symbols remain to be analysed:
plus, PERMUTE, ack, ACK, permute

They will be analysed ascendingly in the following order:
ack < PERMUTE
ACK < PERMUTE
ack < ACK
ack < permute
plus < ack

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n3980_19)))) -> *13_19, rt in Omega(0)

Induction Base:
plus(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, 0))))

Induction Step:
plus(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, +(n3980_19, 1))))) ->_R^Omega(0)
s(plus(s(gen_false:true:0':s9_19(a)), gen_false:true:0':s9_19(+(2, *(2, n3980_19))))) ->_IH
s(*13_19)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
PERMUTE(true, z0, b) -> c5
PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
P(0') -> c7
P(s(z0)) -> c8
ACK(0', z0) -> c9(PLUS(z0, s(0')))
ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
PLUS(0', z0) -> c12
PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
PLUS(z0, s(0')) -> c15
PLUS(z0, 0') -> c16
ISZERO(0') -> c17
ISZERO(s(z0)) -> c18
double(z0) -> permute(z0, z0, a)
permute(z0, z1, a) -> permute(isZero(z0), z0, b)
permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
permute(true, z0, b) -> 0'
permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
p(0') -> 0'
p(s(z0)) -> z0
ack(0', z0) -> plus(z0, s(0'))
ack(s(z0), 0') -> ack(z0, s(0'))
ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
plus(z0, s(0')) -> s(z0)
plus(z0, 0') -> z0
isZero(0') -> true
isZero(s(z0)) -> false

Types:
DOUBLE :: false:true:0':s -> c1
c1 :: c2:c3:c4:c5:c6 -> c1
PERMUTE :: false:true:0':s -> false:true:0':s -> a:b:c -> c2:c3:c4:c5:c6
a :: a:b:c
c2 :: c2:c3:c4:c5:c6 -> c17:c18 -> c2:c3:c4:c5:c6
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
ISZERO :: false:true:0':s -> c17:c18
false :: false:true:0':s
c3 :: c2:c3:c4:c5:c6 -> c9:c10:c11 -> c2:c3:c4:c5:c6
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
ACK :: false:true:0':s -> false:true:0':s -> c9:c10:c11
c4 :: c2:c3:c4:c5:c6 -> c7:c8 -> c2:c3:c4:c5:c6
P :: false:true:0':s -> c7:c8
true :: false:true:0':s
c5 :: c2:c3:c4:c5:c6
c6 :: c2:c3:c4:c5:c6 -> c2:c3:c4:c5:c6
0' :: false:true:0':s
c7 :: c7:c8
s :: false:true:0':s -> false:true:0':s
c8 :: c7:c8
c9 :: c12:c13:c14:c15:c16 -> c9:c10:c11
PLUS :: false:true:0':s -> false:true:0':s -> c12:c13:c14:c15:c16
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14:c15:c16
c13 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c14 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c15 :: c12:c13:c14:c15:c16
c16 :: c12:c13:c14:c15:c16
c17 :: c17:c18
c18 :: c17:c18
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_c11_19 :: c1
hole_false:true:0':s2_19 :: false:true:0':s
hole_c2:c3:c4:c5:c63_19 :: c2:c3:c4:c5:c6
hole_a:b:c4_19 :: a:b:c
hole_c17:c185_19 :: c17:c18
hole_c9:c10:c116_19 :: c9:c10:c11
hole_c7:c87_19 :: c7:c8
hole_c12:c13:c14:c15:c168_19 :: c12:c13:c14:c15:c16
gen_false:true:0':s9_19 :: Nat -> false:true:0':s
gen_c2:c3:c4:c5:c610_19 :: Nat -> c2:c3:c4:c5:c6
gen_c9:c10:c1111_19 :: Nat -> c9:c10:c11
gen_c12:c13:c14:c15:c1612_19 :: Nat -> c12:c13:c14:c15:c16


Lemmas:
PLUS(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n14_19)))) -> *13_19, rt in Omega(n14_19)
plus(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n3980_19)))) -> *13_19, rt in Omega(0)


Generator Equations:
gen_false:true:0':s9_19(0) <=> true
gen_false:true:0':s9_19(+(x, 1)) <=> s(gen_false:true:0':s9_19(x))
gen_c2:c3:c4:c5:c610_19(0) <=> c5
gen_c2:c3:c4:c5:c610_19(+(x, 1)) <=> c2(gen_c2:c3:c4:c5:c610_19(x), c17)
gen_c9:c10:c1111_19(0) <=> c9(c12)
gen_c9:c10:c1111_19(+(x, 1)) <=> c10(gen_c9:c10:c1111_19(x))
gen_c12:c13:c14:c15:c1612_19(0) <=> c12
gen_c12:c13:c14:c15:c1612_19(+(x, 1)) <=> c13(gen_c12:c13:c14:c15:c1612_19(x))


The following defined symbols remain to be analysed:
ack, PERMUTE, ACK, permute

They will be analysed ascendingly in the following order:
ack < PERMUTE
ACK < PERMUTE
ack < ACK
ack < permute

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_false:true:0':s9_19(1), gen_false:true:0':s9_19(+(1, n10639_19))) -> *13_19, rt in Omega(0)

Induction Base:
ack(gen_false:true:0':s9_19(1), gen_false:true:0':s9_19(+(1, 0)))

Induction Step:
ack(gen_false:true:0':s9_19(1), gen_false:true:0':s9_19(+(1, +(n10639_19, 1)))) ->_R^Omega(0)
ack(gen_false:true:0':s9_19(0), ack(s(gen_false:true:0':s9_19(0)), gen_false:true:0':s9_19(+(1, n10639_19)))) ->_IH
ack(gen_false:true:0':s9_19(0), *13_19)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
PERMUTE(true, z0, b) -> c5
PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
P(0') -> c7
P(s(z0)) -> c8
ACK(0', z0) -> c9(PLUS(z0, s(0')))
ACK(s(z0), 0') -> c10(ACK(z0, s(0')))
ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
PLUS(0', z0) -> c12
PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
PLUS(z0, s(0')) -> c15
PLUS(z0, 0') -> c16
ISZERO(0') -> c17
ISZERO(s(z0)) -> c18
double(z0) -> permute(z0, z0, a)
permute(z0, z1, a) -> permute(isZero(z0), z0, b)
permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
permute(true, z0, b) -> 0'
permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
p(0') -> 0'
p(s(z0)) -> z0
ack(0', z0) -> plus(z0, s(0'))
ack(s(z0), 0') -> ack(z0, s(0'))
ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
plus(z0, s(0')) -> s(z0)
plus(z0, 0') -> z0
isZero(0') -> true
isZero(s(z0)) -> false

Types:
DOUBLE :: false:true:0':s -> c1
c1 :: c2:c3:c4:c5:c6 -> c1
PERMUTE :: false:true:0':s -> false:true:0':s -> a:b:c -> c2:c3:c4:c5:c6
a :: a:b:c
c2 :: c2:c3:c4:c5:c6 -> c17:c18 -> c2:c3:c4:c5:c6
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
ISZERO :: false:true:0':s -> c17:c18
false :: false:true:0':s
c3 :: c2:c3:c4:c5:c6 -> c9:c10:c11 -> c2:c3:c4:c5:c6
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
ACK :: false:true:0':s -> false:true:0':s -> c9:c10:c11
c4 :: c2:c3:c4:c5:c6 -> c7:c8 -> c2:c3:c4:c5:c6
P :: false:true:0':s -> c7:c8
true :: false:true:0':s
c5 :: c2:c3:c4:c5:c6
c6 :: c2:c3:c4:c5:c6 -> c2:c3:c4:c5:c6
0' :: false:true:0':s
c7 :: c7:c8
s :: false:true:0':s -> false:true:0':s
c8 :: c7:c8
c9 :: c12:c13:c14:c15:c16 -> c9:c10:c11
PLUS :: false:true:0':s -> false:true:0':s -> c12:c13:c14:c15:c16
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c12 :: c12:c13:c14:c15:c16
c13 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c14 :: c12:c13:c14:c15:c16 -> c12:c13:c14:c15:c16
c15 :: c12:c13:c14:c15:c16
c16 :: c12:c13:c14:c15:c16
c17 :: c17:c18
c18 :: c17:c18
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_c11_19 :: c1
hole_false:true:0':s2_19 :: false:true:0':s
hole_c2:c3:c4:c5:c63_19 :: c2:c3:c4:c5:c6
hole_a:b:c4_19 :: a:b:c
hole_c17:c185_19 :: c17:c18
hole_c9:c10:c116_19 :: c9:c10:c11
hole_c7:c87_19 :: c7:c8
hole_c12:c13:c14:c15:c168_19 :: c12:c13:c14:c15:c16
gen_false:true:0':s9_19 :: Nat -> false:true:0':s
gen_c2:c3:c4:c5:c610_19 :: Nat -> c2:c3:c4:c5:c6
gen_c9:c10:c1111_19 :: Nat -> c9:c10:c11
gen_c12:c13:c14:c15:c1612_19 :: Nat -> c12:c13:c14:c15:c16


Lemmas:
PLUS(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n14_19)))) -> *13_19, rt in Omega(n14_19)
plus(gen_false:true:0':s9_19(a), gen_false:true:0':s9_19(+(2, *(2, n3980_19)))) -> *13_19, rt in Omega(0)
ack(gen_false:true:0':s9_19(1), gen_false:true:0':s9_19(+(1, n10639_19))) -> *13_19, rt in Omega(0)


Generator Equations:
gen_false:true:0':s9_19(0) <=> true
gen_false:true:0':s9_19(+(x, 1)) <=> s(gen_false:true:0':s9_19(x))
gen_c2:c3:c4:c5:c610_19(0) <=> c5
gen_c2:c3:c4:c5:c610_19(+(x, 1)) <=> c2(gen_c2:c3:c4:c5:c610_19(x), c17)
gen_c9:c10:c1111_19(0) <=> c9(c12)
gen_c9:c10:c1111_19(+(x, 1)) <=> c10(gen_c9:c10:c1111_19(x))
gen_c12:c13:c14:c15:c1612_19(0) <=> c12
gen_c12:c13:c14:c15:c1612_19(+(x, 1)) <=> c13(gen_c12:c13:c14:c15:c1612_19(x))


The following defined symbols remain to be analysed:
ACK, PERMUTE, permute

They will be analysed ascendingly in the following order:
ACK < PERMUTE

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

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

(26)
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:

   double(x) -> permute(x, x, a) [1]
   permute(x, y, a) -> permute(isZero(x), x, b) [1]
   permute(false, x, b) -> permute(ack(x, x), p(x), c) [1]
   permute(true, x, b) -> 0 [1]
   permute(y, x, c) -> s(s(permute(x, y, a))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ack(0, x) -> plus(x, s(0)) [1]
   ack(s(x), 0) -> ack(x, s(0)) [1]
   ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   plus(x, s(s(y))) -> s(plus(s(x), y)) [1]
   plus(x, s(0)) -> s(x) [1]
   plus(x, 0) -> x [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]

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

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

(28)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   double(x) -> permute(x, x, a) [1]
   permute(x, y, a) -> permute(isZero(x), x, b) [1]
   permute(false, x, b) -> permute(ack(x, x), p(x), c) [1]
   permute(true, x, b) -> 0 [1]
   permute(y, x, c) -> s(s(permute(x, y, a))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ack(0, x) -> plus(x, s(0)) [1]
   ack(s(x), 0) -> ack(x, s(0)) [1]
   ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   plus(x, s(s(y))) -> s(plus(s(x), y)) [1]
   plus(x, s(0)) -> s(x) [1]
   plus(x, 0) -> x [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]

The TRS has the following type information:
double :: false:true:0:s -> false:true:0:s
permute :: false:true:0:s -> false:true:0:s -> a:b:c -> false:true:0:s
a :: a:b:c
isZero :: false:true:0:s -> false:true:0:s
b :: a:b:c
false :: false:true:0:s
ack :: false:true:0:s -> false:true:0:s -> false:true:0:s
p :: false:true:0:s -> false:true:0:s
c :: a:b:c
true :: false:true:0:s
0 :: false:true:0:s
s :: false:true:0:s -> false:true:0:s
plus :: false:true:0:s -> false:true:0:s -> false:true:0:s

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

(29) 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:

   double_1
   permute_3

(c) The following functions are completely defined:

   isZero_1
   ack_2
   p_1
   plus_2

Due to the following rules being added:

   isZero(v0) -> null_isZero [0]
   ack(v0, v1) -> null_ack [0]
   p(v0) -> null_p [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_isZero, null_ack, null_p, null_plus

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

(30)
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:

   double(x) -> permute(x, x, a) [1]
   permute(x, y, a) -> permute(isZero(x), x, b) [1]
   permute(false, x, b) -> permute(ack(x, x), p(x), c) [1]
   permute(true, x, b) -> 0 [1]
   permute(y, x, c) -> s(s(permute(x, y, a))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ack(0, x) -> plus(x, s(0)) [1]
   ack(s(x), 0) -> ack(x, s(0)) [1]
   ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   plus(x, s(s(y))) -> s(plus(s(x), y)) [1]
   plus(x, s(0)) -> s(x) [1]
   plus(x, 0) -> x [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   isZero(v0) -> null_isZero [0]
   ack(v0, v1) -> null_ack [0]
   p(v0) -> null_p [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
double :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
permute :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus -> a:b:c -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
a :: a:b:c
isZero :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
b :: a:b:c
false :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
ack :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
p :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
c :: a:b:c
true :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
0 :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
s :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
plus :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_isZero :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_ack :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_p :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_plus :: false:true:0:s:null_isZero:null_ack:null_p:null_plus

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

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

(32)
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:

   double(x) -> permute(x, x, a) [1]
   permute(0, y, a) -> permute(true, 0, b) [2]
   permute(s(x'), y, a) -> permute(false, s(x'), b) [2]
   permute(x, y, a) -> permute(null_isZero, x, b) [1]
   permute(false, 0, b) -> permute(plus(0, s(0)), 0, c) [3]
   permute(false, 0, b) -> permute(plus(0, s(0)), null_p, c) [2]
   permute(false, s(x''), b) -> permute(ack(x'', ack(s(x''), x'')), x'', c) [3]
   permute(false, s(x''), b) -> permute(ack(x'', ack(s(x''), x'')), null_p, c) [2]
   permute(false, 0, b) -> permute(null_ack, 0, c) [2]
   permute(false, s(x1), b) -> permute(null_ack, x1, c) [2]
   permute(false, x, b) -> permute(null_ack, null_p, c) [1]
   permute(true, x, b) -> 0 [1]
   permute(y, x, c) -> s(s(permute(x, y, a))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ack(0, x) -> plus(x, s(0)) [1]
   ack(s(x), 0) -> ack(x, s(0)) [1]
   ack(s(x), s(0)) -> ack(x, ack(x, s(0))) [2]
   ack(s(x), s(s(y'))) -> ack(x, ack(x, ack(s(x), y'))) [2]
   ack(s(x), s(y)) -> ack(x, null_ack) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   plus(x, s(s(y))) -> s(plus(s(x), y)) [1]
   plus(x, s(0)) -> s(x) [1]
   plus(x, 0) -> x [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   isZero(v0) -> null_isZero [0]
   ack(v0, v1) -> null_ack [0]
   p(v0) -> null_p [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
double :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
permute :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus -> a:b:c -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
a :: a:b:c
isZero :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
b :: a:b:c
false :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
ack :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
p :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
c :: a:b:c
true :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
0 :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
s :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
plus :: false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus -> false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_isZero :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_ack :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_p :: false:true:0:s:null_isZero:null_ack:null_p:null_plus
null_plus :: false:true:0:s:null_isZero:null_ack:null_p:null_plus

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

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

   a => 0
   b => 1
   false => 1
   c => 2
   true => 2
   0 => 0
   null_isZero => 0
   null_ack => 0
   null_p => 0
   null_plus => 0

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

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

   ack(z, z') -{ 1 }-> plus(x, 1 + 0) :|: z' = x, x >= 0, z = 0
   ack(z, z') -{ 2 }-> ack(x, ack(x, ack(1 + x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = 1 + x
   ack(z, z') -{ 2 }-> ack(x, ack(x, 1 + 0)) :|: x >= 0, z' = 1 + 0, z = 1 + x
   ack(z, z') -{ 1 }-> ack(x, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ack(z, z') -{ 1 }-> ack(x, 1 + 0) :|: x >= 0, z = 1 + x, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   double(z) -{ 1 }-> permute(x, x, 0) :|: x >= 0, z = x
   isZero(z) -{ 1 }-> 2 :|: z = 0
   isZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   isZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   permute(z, z', z'') -{ 3 }-> permute(plus(0, 1 + 0), 0, 2) :|: z = 1, z'' = 1, z' = 0
   permute(z, z', z'') -{ 2 }-> permute(plus(0, 1 + 0), 0, 2) :|: z = 1, z'' = 1, z' = 0
   permute(z, z', z'') -{ 3 }-> permute(ack(x'', ack(1 + x'', x'')), x'', 2) :|: z' = 1 + x'', z = 1, x'' >= 0, z'' = 1
   permute(z, z', z'') -{ 2 }-> permute(ack(x'', ack(1 + x'', x'')), 0, 2) :|: z' = 1 + x'', z = 1, x'' >= 0, z'' = 1
   permute(z, z', z'') -{ 2 }-> permute(2, 0, 1) :|: z'' = 0, y >= 0, z = 0, z' = y
   permute(z, z', z'') -{ 2 }-> permute(1, 1 + x', 1) :|: z = 1 + x', z'' = 0, x' >= 0, y >= 0, z' = y
   permute(z, z', z'') -{ 1 }-> permute(0, x, 1) :|: z'' = 0, x >= 0, y >= 0, z = x, z' = y
   permute(z, z', z'') -{ 2 }-> permute(0, x1, 2) :|: x1 >= 0, z = 1, z' = 1 + x1, z'' = 1
   permute(z, z', z'') -{ 2 }-> permute(0, 0, 2) :|: z = 1, z'' = 1, z' = 0
   permute(z, z', z'') -{ 1 }-> permute(0, 0, 2) :|: z' = x, z = 1, x >= 0, z'' = 1
   permute(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, x >= 0, z'' = 1
   permute(z, z', z'') -{ 1 }-> 1 + (1 + permute(x, y, 0)) :|: z' = x, y >= 0, x >= 0, z'' = 2, z = y
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> plus(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   plus(z, z') -{ 1 }-> 1 + plus(1 + x, y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x


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

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

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

   ack(z, z') -{ 1 }-> plus(z', 1 + 0) :|: z' >= 0, z = 0
   ack(z, z') -{ 2 }-> ack(z - 1, ack(z - 1, ack(1 + (z - 1), z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0
   ack(z, z') -{ 2 }-> ack(z - 1, ack(z - 1, 1 + 0)) :|: z - 1 >= 0, z' = 1 + 0
   ack(z, z') -{ 1 }-> ack(z - 1, 0) :|: z - 1 >= 0, z' - 1 >= 0
   ack(z, z') -{ 1 }-> ack(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   double(z) -{ 1 }-> permute(z, z, 0) :|: z >= 0
   isZero(z) -{ 1 }-> 2 :|: z = 0
   isZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   isZero(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   permute(z, z', z'') -{ 3 }-> permute(plus(0, 1 + 0), 0, 2) :|: z = 1, z'' = 1, z' = 0
   permute(z, z', z'') -{ 2 }-> permute(plus(0, 1 + 0), 0, 2) :|: z = 1, z'' = 1, z' = 0
   permute(z, z', z'') -{ 2 }-> permute(ack(z' - 1, ack(1 + (z' - 1), z' - 1)), 0, 2) :|: z = 1, z' - 1 >= 0, z'' = 1
   permute(z, z', z'') -{ 3 }-> permute(ack(z' - 1, ack(1 + (z' - 1), z' - 1)), z' - 1, 2) :|: z = 1, z' - 1 >= 0, z'' = 1
   permute(z, z', z'') -{ 2 }-> permute(2, 0, 1) :|: z'' = 0, z' >= 0, z = 0
   permute(z, z', z'') -{ 2 }-> permute(1, 1 + (z - 1), 1) :|: z'' = 0, z - 1 >= 0, z' >= 0
   permute(z, z', z'') -{ 1 }-> permute(0, z, 1) :|: z'' = 0, z >= 0, z' >= 0
   permute(z, z', z'') -{ 2 }-> permute(0, 0, 2) :|: z = 1, z'' = 1, z' = 0
   permute(z, z', z'') -{ 1 }-> permute(0, 0, 2) :|: z = 1, z' >= 0, z'' = 1
   permute(z, z', z'') -{ 2 }-> permute(0, z' - 1, 2) :|: z' - 1 >= 0, z = 1, z'' = 1
   permute(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' = 1
   permute(z, z', z'') -{ 1 }-> 1 + (1 + permute(z', z, 0)) :|: z >= 0, z' >= 0, z'' = 2
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 1 }-> 1 + plus(1 + z, z' - 2) :|: z >= 0, z' - 2 >= 0


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

(37) 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:

   permute(v0, v1, v2) -> null_permute [0]
   p(v0) -> null_p [0]
   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   isZero(v0) -> null_isZero [0]

And the following fresh constants:    null_permute, null_p, null_ack, null_plus, null_isZero

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

(38)
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:

   double(x) -> permute(x, x, a) [1]
   permute(x, y, a) -> permute(isZero(x), x, b) [1]
   permute(false, x, b) -> permute(ack(x, x), p(x), c) [1]
   permute(true, x, b) -> 0 [1]
   permute(y, x, c) -> s(s(permute(x, y, a))) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ack(0, x) -> plus(x, s(0)) [1]
   ack(s(x), 0) -> ack(x, s(0)) [1]
   ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   plus(x, s(s(y))) -> s(plus(s(x), y)) [1]
   plus(x, s(0)) -> s(x) [1]
   plus(x, 0) -> x [1]
   isZero(0) -> true [1]
   isZero(s(x)) -> false [1]
   permute(v0, v1, v2) -> null_permute [0]
   p(v0) -> null_p [0]
   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   isZero(v0) -> null_isZero [0]

The TRS has the following type information:
double :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
permute :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> a:b:c -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
a :: a:b:c
isZero :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
b :: a:b:c
false :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
ack :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
p :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
c :: a:b:c
true :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
0 :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
s :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
plus :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero -> false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
null_permute :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
null_p :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
null_ack :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
null_plus :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero
null_isZero :: false:true:0:s:null_permute:null_p:null_ack:null_plus:null_isZero

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

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

   a => 0
   b => 1
   false => 1
   c => 2
   true => 2
   0 => 0
   null_permute => 0
   null_p => 0
   null_ack => 0
   null_plus => 0
   null_isZero => 0

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

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

   ack(z, z') -{ 1 }-> plus(x, 1 + 0) :|: z' = x, x >= 0, z = 0
   ack(z, z') -{ 1 }-> ack(x, ack(1 + x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ack(z, z') -{ 1 }-> ack(x, 1 + 0) :|: x >= 0, z = 1 + x, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   double(z) -{ 1 }-> permute(x, x, 0) :|: x >= 0, z = x
   isZero(z) -{ 1 }-> 2 :|: z = 0
   isZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   isZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   permute(z, z', z'') -{ 1 }-> permute(isZero(x), x, 1) :|: z'' = 0, x >= 0, y >= 0, z = x, z' = y
   permute(z, z', z'') -{ 1 }-> permute(ack(x, x), p(x), 2) :|: z' = x, z = 1, x >= 0, z'' = 1
   permute(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, x >= 0, z'' = 1
   permute(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   permute(z, z', z'') -{ 1 }-> 1 + (1 + permute(x, y, 0)) :|: z' = x, y >= 0, x >= 0, z'' = 2, z = y
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> plus(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   plus(z, z') -{ 1 }-> 1 + plus(1 + x, y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18
S tuples:
   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18
K tuples:none
Defined Rule Symbols:   double_1, permute_3, p_1, ack_2, plus_2, isZero_1

Defined Pair Symbols:   DOUBLE_1, PERMUTE_3, P_1, ACK_2, PLUS_2, ISZERO_1

Compound Symbols:   c1_1, c2_2, c3_2, c4_2, c5, c6_1, c7, c8, c9_1, c10_1, c11_2, c12, c13_1, c14_1, c15, c16, c17, c18


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
Removed 8 trailing nodes:
   ISZERO(0) -> c17
   P(s(z0)) -> c8
   PERMUTE(true, z0, b) -> c5
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   PLUS(0, z0) -> c12
   P(0) -> c7
   ISZERO(s(z0)) -> c18

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
S tuples:
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
K tuples:none
Defined Rule Symbols:   double_1, permute_3, p_1, ack_2, plus_2, isZero_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c2_2, c3_2, c4_2, c6_1, c9_1, c10_1, c11_2, c13_1, c14_1


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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
S tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
K tuples:none
Defined Rule Symbols:   double_1, permute_3, p_1, ack_2, plus_2, isZero_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c3_2, c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c4_1


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

(47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
S tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1, isZero_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c3_2, c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c4_1


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

(49) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(50)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))

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

   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

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

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

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

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

(53) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(54)
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:

   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0)) [1]
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a)) [1]
   ACK(0, z0) -> c9(PLUS(z0, s(0))) [1]
   ACK(s(z0), 0) -> c10(ACK(z0, s(0))) [1]
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) [1]
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1))) [1]
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1)) [1]
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b)) [1]
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c)) [1]
   ack(0, z0) -> plus(z0, s(0)) [0]
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) [0]
   ack(s(z0), 0) -> ack(z0, s(0)) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   plus(z0, s(0)) -> s(z0) [0]
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) [0]
   plus(z0, 0) -> z0 [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   isZero(0) -> true [0]
   isZero(s(z0)) -> false [0]

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

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

(56)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0)) [1]
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a)) [1]
   ACK(0, z0) -> c9(PLUS(z0, s(0))) [1]
   ACK(s(z0), 0) -> c10(ACK(z0, s(0))) [1]
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) [1]
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1))) [1]
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1)) [1]
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b)) [1]
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c)) [1]
   ack(0, z0) -> plus(z0, s(0)) [0]
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) [0]
   ack(s(z0), 0) -> ack(z0, s(0)) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   plus(z0, s(0)) -> s(z0) [0]
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) [0]
   plus(z0, 0) -> z0 [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   isZero(0) -> true [0]
   isZero(s(z0)) -> false [0]

The TRS has the following type information:
PERMUTE :: false:0:s:true -> false:0:s:true -> b:c:a -> c3:c6:c2:c4
false :: false:0:s:true
b :: b:c:a
c3 :: c3:c6:c2:c4 -> c9:c10:c11 -> c3:c6:c2:c4
ack :: false:0:s:true -> false:0:s:true -> false:0:s:true
p :: false:0:s:true -> false:0:s:true
c :: b:c:a
ACK :: false:0:s:true -> false:0:s:true -> c9:c10:c11
c6 :: c3:c6:c2:c4 -> c3:c6:c2:c4
a :: b:c:a
0 :: false:0:s:true
c9 :: c13:c14 -> c9:c10:c11
PLUS :: false:0:s:true -> false:0:s:true -> c13:c14
s :: false:0:s:true -> false:0:s:true
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c13 :: c13:c14 -> c13:c14
c14 :: c13:c14 -> c13:c14
c2 :: c3:c6:c2:c4 -> c3:c6:c2:c4
isZero :: false:0:s:true -> false:0:s:true
c4 :: c3:c6:c2:c4 -> c3:c6:c2:c4
plus :: false:0:s:true -> false:0:s:true -> false:0:s:true
true :: false:0:s:true

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

(57) 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:

   PERMUTE_3
   ACK_2
   PLUS_2

(c) The following functions are completely defined:

   ack_2
   plus_2
   p_1
   isZero_1

Due to the following rules being added:

   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]

And the following fresh constants:    null_ack, null_plus, null_p, null_isZero, const, const1, const2

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

(58)
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:

   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0)) [1]
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a)) [1]
   ACK(0, z0) -> c9(PLUS(z0, s(0))) [1]
   ACK(s(z0), 0) -> c10(ACK(z0, s(0))) [1]
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) [1]
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1))) [1]
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1)) [1]
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b)) [1]
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c)) [1]
   ack(0, z0) -> plus(z0, s(0)) [0]
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) [0]
   ack(s(z0), 0) -> ack(z0, s(0)) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   plus(z0, s(0)) -> s(z0) [0]
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) [0]
   plus(z0, 0) -> z0 [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   isZero(0) -> true [0]
   isZero(s(z0)) -> false [0]
   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]

The TRS has the following type information:
PERMUTE :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> b:c:a -> c3:c6:c2:c4
false :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
b :: b:c:a
c3 :: c3:c6:c2:c4 -> c9:c10:c11 -> c3:c6:c2:c4
ack :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
p :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c :: b:c:a
ACK :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> c9:c10:c11
c6 :: c3:c6:c2:c4 -> c3:c6:c2:c4
a :: b:c:a
0 :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
c9 :: c13:c14 -> c9:c10:c11
PLUS :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> c13:c14
s :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c13 :: c13:c14 -> c13:c14
c14 :: c13:c14 -> c13:c14
c2 :: c3:c6:c2:c4 -> c3:c6:c2:c4
isZero :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c4 :: c3:c6:c2:c4 -> c3:c6:c2:c4
plus :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
true :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_ack :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_plus :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_p :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_isZero :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
const :: c3:c6:c2:c4
const1 :: c9:c10:c11
const2 :: c13:c14

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

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

(60)
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:

   PERMUTE(false, 0, b) -> c3(PERMUTE(plus(0, s(0)), 0, c), ACK(0, 0)) [1]
   PERMUTE(false, 0, b) -> c3(PERMUTE(plus(0, s(0)), null_p, c), ACK(0, 0)) [1]
   PERMUTE(false, s(z0'), b) -> c3(PERMUTE(ack(z0', ack(s(z0'), z0')), z0', c), ACK(s(z0'), s(z0'))) [1]
   PERMUTE(false, s(z0'), b) -> c3(PERMUTE(ack(z0', ack(s(z0'), z0')), null_p, c), ACK(s(z0'), s(z0'))) [1]
   PERMUTE(false, 0, b) -> c3(PERMUTE(null_ack, 0, c), ACK(0, 0)) [1]
   PERMUTE(false, s(z0''), b) -> c3(PERMUTE(null_ack, z0'', c), ACK(s(z0''), s(z0''))) [1]
   PERMUTE(false, z0, b) -> c3(PERMUTE(null_ack, null_p, c), ACK(z0, z0)) [1]
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a)) [1]
   ACK(0, z0) -> c9(PLUS(z0, s(0))) [1]
   ACK(s(z0), 0) -> c10(ACK(z0, s(0))) [1]
   ACK(s(z0), s(s(z1'))) -> c11(ACK(z0, ack(z0, ack(s(z0), z1'))), ACK(s(z0), s(z1'))) [1]
   ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) [1]
   ACK(s(z0), s(z1)) -> c11(ACK(z0, null_ack), ACK(s(z0), z1)) [1]
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1))) [1]
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1)) [1]
   PERMUTE(0, z1, a) -> c2(PERMUTE(true, 0, b)) [1]
   PERMUTE(s(z01), z1, a) -> c2(PERMUTE(false, s(z01), b)) [1]
   PERMUTE(z0, z1, a) -> c2(PERMUTE(null_isZero, z0, b)) [1]
   PERMUTE(false, 0, b) -> c4(PERMUTE(plus(0, s(0)), 0, c)) [1]
   PERMUTE(false, 0, b) -> c4(PERMUTE(plus(0, s(0)), null_p, c)) [1]
   PERMUTE(false, s(z02), b) -> c4(PERMUTE(ack(z02, ack(s(z02), z02)), z02, c)) [1]
   PERMUTE(false, s(z02), b) -> c4(PERMUTE(ack(z02, ack(s(z02), z02)), null_p, c)) [1]
   PERMUTE(false, 0, b) -> c4(PERMUTE(null_ack, 0, c)) [1]
   PERMUTE(false, s(z03), b) -> c4(PERMUTE(null_ack, z03, c)) [1]
   PERMUTE(false, z0, b) -> c4(PERMUTE(null_ack, null_p, c)) [1]
   ack(0, z0) -> plus(z0, s(0)) [0]
   ack(s(z0), s(s(z1''))) -> ack(z0, ack(z0, ack(s(z0), z1''))) [0]
   ack(s(z0), s(0)) -> ack(z0, ack(z0, s(0))) [0]
   ack(s(z0), s(z1)) -> ack(z0, null_ack) [0]
   ack(s(z0), 0) -> ack(z0, s(0)) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   plus(z0, s(0)) -> s(z0) [0]
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) [0]
   plus(z0, 0) -> z0 [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   isZero(0) -> true [0]
   isZero(s(z0)) -> false [0]
   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]

The TRS has the following type information:
PERMUTE :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> b:c:a -> c3:c6:c2:c4
false :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
b :: b:c:a
c3 :: c3:c6:c2:c4 -> c9:c10:c11 -> c3:c6:c2:c4
ack :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
p :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c :: b:c:a
ACK :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> c9:c10:c11
c6 :: c3:c6:c2:c4 -> c3:c6:c2:c4
a :: b:c:a
0 :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
c9 :: c13:c14 -> c9:c10:c11
PLUS :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> c13:c14
s :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c10 :: c9:c10:c11 -> c9:c10:c11
c11 :: c9:c10:c11 -> c9:c10:c11 -> c9:c10:c11
c13 :: c13:c14 -> c13:c14
c14 :: c13:c14 -> c13:c14
c2 :: c3:c6:c2:c4 -> c3:c6:c2:c4
isZero :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c4 :: c3:c6:c2:c4 -> c3:c6:c2:c4
plus :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
true :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_ack :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_plus :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_p :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_isZero :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
const :: c3:c6:c2:c4
const1 :: c9:c10:c11
const2 :: c13:c14

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

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

   false => 1
   b => 1
   c => 2
   a => 0
   0 => 0
   true => 2
   null_ack => 0
   null_plus => 0
   null_p => 0
   null_isZero => 0
   const => 0
   const1 => 0
   const2 => 0

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

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

   ACK(z, z') -{ 1 }-> 1 + PLUS(z0, 1 + 0) :|: z0 >= 0, z = 0, z' = z0
   ACK(z, z') -{ 1 }-> 1 + ACK(z0, 1 + 0) :|: z = 1 + z0, z0 >= 0, z' = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z0, ack(z0, ack(1 + z0, z1'))) + ACK(1 + z0, 1 + z1') :|: z' = 1 + (1 + z1'), z = 1 + z0, z1' >= 0, z0 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z0, ack(z0, 1 + 0)) + ACK(1 + z0, 0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z0, 0) + ACK(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(z1, z0, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 2
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(plus(0, 1 + 0), 0, 2) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z02, ack(1 + z02, z02)), z02, 2) :|: z = 1, z02 >= 0, z' = 1 + z02, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z02, ack(1 + z02, z02)), 0, 2) :|: z = 1, z02 >= 0, z' = 1 + z02, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(2, 0, 1) :|: z'' = 0, z1 >= 0, z' = z1, z = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(1, 1 + z01, 1) :|: z'' = 0, z1 >= 0, z01 >= 0, z = 1 + z01, z' = z1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, z0, 1) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, z03, 2) :|: z = 1, z' = 1 + z03, z03 >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) :|: z = 1, z0 >= 0, z' = z0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(plus(0, 1 + 0), 0, 2) + ACK(0, 0) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z0', ack(1 + z0', z0')), z0', 2) + ACK(1 + z0', 1 + z0') :|: z = 1, z0' >= 0, z' = 1 + z0', z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z0', ack(1 + z0', z0')), 0, 2) + ACK(1 + z0', 1 + z0') :|: z = 1, z0' >= 0, z' = 1 + z0', z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, z0'', 2) + ACK(1 + z0'', 1 + z0'') :|: z = 1, z' = 1 + z0'', z0'' >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) + ACK(z0, z0) :|: z = 1, z0 >= 0, z' = z0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) + ACK(0, 0) :|: z = 1, z'' = 1, z' = 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(1 + z0, z1) :|: z' = 1 + (1 + z1), z = z0, z1 >= 0, z0 >= 0
   ack(z, z') -{ 0 }-> plus(z0, 1 + 0) :|: z0 >= 0, z = 0, z' = z0
   ack(z, z') -{ 0 }-> ack(z0, ack(z0, ack(1 + z0, z1''))) :|: z' = 1 + (1 + z1''), z = 1 + z0, z0 >= 0, z1'' >= 0
   ack(z, z') -{ 0 }-> ack(z0, ack(z0, 1 + 0)) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0
   ack(z, z') -{ 0 }-> ack(z0, 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ack(z, z') -{ 0 }-> ack(z0, 1 + 0) :|: z = 1 + z0, z0 >= 0, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   isZero(z) -{ 0 }-> 2 :|: z = 0
   isZero(z) -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0
   isZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> plus(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + z0 :|: z = z0, z' = 1 + 0, z0 >= 0
   plus(z, z') -{ 0 }-> 1 + plus(1 + z0, z1) :|: z' = 1 + (1 + z1), z = z0, z1 >= 0, z0 >= 0


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

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

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

   ACK(z, z') -{ 1 }-> 1 + PLUS(z', 1 + 0) :|: z' >= 0, z = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z - 1, ack(z - 1, ack(1 + (z - 1), z' - 2))) + ACK(1 + (z - 1), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 1 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z - 1, ack(z - 1, 1 + 0)) + ACK(1 + (z - 1), 0) :|: z' = 1 + 0, z - 1 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z - 1, 0) + ACK(1 + (z - 1), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(z', z, 0) :|: z' >= 0, z >= 0, z'' = 2
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(plus(0, 1 + 0), 0, 2) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z' - 1, ack(1 + (z' - 1), z' - 1)), 0, 2) :|: z = 1, z' - 1 >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z' - 1, ack(1 + (z' - 1), z' - 1)), z' - 1, 2) :|: z = 1, z' - 1 >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(2, 0, 1) :|: z'' = 0, z' >= 0, z = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(1, 1 + (z - 1), 1) :|: z'' = 0, z' >= 0, z - 1 >= 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, z, 1) :|: z'' = 0, z' >= 0, z >= 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) :|: z = 1, z' >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, z' - 1, 2) :|: z = 1, z' - 1 >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(plus(0, 1 + 0), 0, 2) + ACK(0, 0) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z' - 1, ack(1 + (z' - 1), z' - 1)), 0, 2) + ACK(1 + (z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z' - 1, ack(1 + (z' - 1), z' - 1)), z' - 1, 2) + ACK(1 + (z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) + ACK(z', z') :|: z = 1, z' >= 0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, 0, 2) + ACK(0, 0) :|: z = 1, z'' = 1, z' = 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(0, z' - 1, 2) + ACK(1 + (z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0
   ack(z, z') -{ 0 }-> plus(z', 1 + 0) :|: z' >= 0, z = 0
   ack(z, z') -{ 0 }-> ack(z - 1, ack(z - 1, ack(1 + (z - 1), z' - 2))) :|: z - 1 >= 0, z' - 2 >= 0
   ack(z, z') -{ 0 }-> ack(z - 1, ack(z - 1, 1 + 0)) :|: z' = 1 + 0, z - 1 >= 0
   ack(z, z') -{ 0 }-> ack(z - 1, 0) :|: z' - 1 >= 0, z - 1 >= 0
   ack(z, z') -{ 0 }-> ack(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   isZero(z) -{ 0 }-> 2 :|: z = 0
   isZero(z) -{ 0 }-> 1 :|: z - 1 >= 0
   isZero(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> plus(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z' = 1 + 0, z >= 0
   plus(z, z') -{ 0 }-> 1 + plus(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0


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

(65) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]
   PERMUTE(v0, v1, v2) -> null_PERMUTE [0]
   ACK(v0, v1) -> null_ACK [0]
   PLUS(v0, v1) -> null_PLUS [0]

And the following fresh constants:    null_ack, null_plus, null_p, null_isZero, null_PERMUTE, null_ACK, null_PLUS

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

(66)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0)) [1]
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a)) [1]
   ACK(0, z0) -> c9(PLUS(z0, s(0))) [1]
   ACK(s(z0), 0) -> c10(ACK(z0, s(0))) [1]
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) [1]
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1))) [1]
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1)) [1]
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b)) [1]
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c)) [1]
   ack(0, z0) -> plus(z0, s(0)) [0]
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) [0]
   ack(s(z0), 0) -> ack(z0, s(0)) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> plus(z0, s(z1)) [0]
   plus(z0, s(0)) -> s(z0) [0]
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1)) [0]
   plus(z0, 0) -> z0 [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   isZero(0) -> true [0]
   isZero(s(z0)) -> false [0]
   ack(v0, v1) -> null_ack [0]
   plus(v0, v1) -> null_plus [0]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]
   PERMUTE(v0, v1, v2) -> null_PERMUTE [0]
   ACK(v0, v1) -> null_ACK [0]
   PLUS(v0, v1) -> null_PLUS [0]

The TRS has the following type information:
PERMUTE :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> b:c:a -> c3:c6:c2:c4:null_PERMUTE
false :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
b :: b:c:a
c3 :: c3:c6:c2:c4:null_PERMUTE -> c9:c10:c11:null_ACK -> c3:c6:c2:c4:null_PERMUTE
ack :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
p :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c :: b:c:a
ACK :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> c9:c10:c11:null_ACK
c6 :: c3:c6:c2:c4:null_PERMUTE -> c3:c6:c2:c4:null_PERMUTE
a :: b:c:a
0 :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
c9 :: c13:c14:null_PLUS -> c9:c10:c11:null_ACK
PLUS :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> c13:c14:null_PLUS
s :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c10 :: c9:c10:c11:null_ACK -> c9:c10:c11:null_ACK
c11 :: c9:c10:c11:null_ACK -> c9:c10:c11:null_ACK -> c9:c10:c11:null_ACK
c13 :: c13:c14:null_PLUS -> c13:c14:null_PLUS
c14 :: c13:c14:null_PLUS -> c13:c14:null_PLUS
c2 :: c3:c6:c2:c4:null_PERMUTE -> c3:c6:c2:c4:null_PERMUTE
isZero :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
c4 :: c3:c6:c2:c4:null_PERMUTE -> c3:c6:c2:c4:null_PERMUTE
plus :: false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero -> false:0:s:true:null_ack:null_plus:null_p:null_isZero
true :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_ack :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_plus :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_p :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_isZero :: false:0:s:true:null_ack:null_plus:null_p:null_isZero
null_PERMUTE :: c3:c6:c2:c4:null_PERMUTE
null_ACK :: c9:c10:c11:null_ACK
null_PLUS :: c13:c14:null_PLUS

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

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

   false => 1
   b => 1
   c => 2
   a => 0
   0 => 0
   true => 2
   null_ack => 0
   null_plus => 0
   null_p => 0
   null_isZero => 0
   null_PERMUTE => 0
   null_ACK => 0
   null_PLUS => 0

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

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

   ACK(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ACK(z, z') -{ 1 }-> 1 + PLUS(z0, 1 + 0) :|: z0 >= 0, z = 0, z' = z0
   ACK(z, z') -{ 1 }-> 1 + ACK(z0, 1 + 0) :|: z = 1 + z0, z0 >= 0, z' = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z0, ack(1 + z0, z1)) + ACK(1 + z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   PERMUTE(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(z1, z0, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 2
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(isZero(z0), z0, 1) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z0, z0), p(z0), 2) :|: z = 1, z0 >= 0, z' = z0, z'' = 1
   PERMUTE(z, z', z'') -{ 1 }-> 1 + PERMUTE(ack(z0, z0), p(z0), 2) + ACK(z0, z0) :|: z = 1, z0 >= 0, z' = z0, z'' = 1
   PLUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(1 + z0, z1) :|: z' = 1 + (1 + z1), z = z0, z1 >= 0, z0 >= 0
   ack(z, z') -{ 0 }-> plus(z0, 1 + 0) :|: z0 >= 0, z = 0, z' = z0
   ack(z, z') -{ 0 }-> ack(z0, ack(1 + z0, z1)) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ack(z, z') -{ 0 }-> ack(z0, 1 + 0) :|: z = 1 + z0, z0 >= 0, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   isZero(z) -{ 0 }-> 2 :|: z = 0
   isZero(z) -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0
   isZero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> plus(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + z0 :|: z = z0, z' = 1 + 0, z0 >= 0
   plus(z, z') -{ 0 }-> 1 + plus(1 + z0, z1) :|: z' = 1 + (1 + z1), z = z0, z1 >= 0, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b)) by 
   PERMUTE(0, x1, a) -> c2(PERMUTE(true, 0, b))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(0, x1, a) -> c2(PERMUTE(true, 0, b))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
S tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(0, x1, a) -> c2(PERMUTE(true, 0, b))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1, isZero_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c3_2, c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PERMUTE(0, x1, a) -> c2(PERMUTE(true, 0, b))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false
Tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
S tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1, isZero_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c3_2, c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1


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

(73) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   isZero(0) -> true
   isZero(s(z0)) -> false

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
S tuples:
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c3_2, c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1


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

(75) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0)) by 
   PERMUTE(false, s(x0), b) -> c3(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c), ACK(s(x0), s(x0)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(x0), b) -> c3(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c), ACK(s(x0), s(x0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(x0), b) -> c3(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c), ACK(s(x0), s(x0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(x0), b) -> c3(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c), ACK(s(x0), s(x0))) by 
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(s(z0), s(z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c), ACK(s(z0), s(z0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(s(z0), s(z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c), ACK(s(z0), s(z0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(s(z0), s(z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c), ACK(s(z0), s(z0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(s(z0), s(z0)), z0, c), ACK(s(z0), s(z0))) by 
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c), ACK(s(z0), s(z0))) by 
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c), ACK(s(s(z1)), s(s(z1))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c), ACK(s(s(z1)), s(s(z1))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c), ACK(s(s(z1)), s(s(z1))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c), ACK(s(0), s(0))) by PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c), ACK(s(s(z1)), s(s(z1))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c), ACK(s(s(z1)), s(s(z1))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c), ACK(s(s(z1)), s(s(z1)))) by PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(87) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) by PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c4_1, c2_1, c3_2


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c)) by 
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c4_1


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

(91) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   p(0) -> 0

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c4_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c4_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c4_1, c3_1


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

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ACK(x_1, x_2)) = 0
   POL(PERMUTE(x_1, x_2, x_3)) = [1] + x_3
   POL(PLUS(x_1, x_2)) = 0
   POL(a) = 0
   POL(ack(x_1, x_2)) = [1]
   POL(b) = 0
   POL(c) = 0
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(false) = [1]
   POL(p(x_1)) = [1] + x_1
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = [1]

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c4_1, c3_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c)) by 
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), 0, c), ACK(s(0), s(0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(105) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), 0, c), ACK(s(0), s(0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), 0, c), ACK(s(0), s(0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(111) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(s(z0), s(z0)), z0, c)) by 
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), p(s(z0)), c)) by 
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
S tuples:
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c6_1, c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a)) by 
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

(119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing tuple parts
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

(121) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0)))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(z0), b) -> c3(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c), ACK(s(z0), s(z0))) by 
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(s(z1)), b) -> c3(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), s(z1), c), ACK(s(s(z1)), s(s(z1))))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

(127) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

(129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ACK(x_1, x_2)) = 0
   POL(PERMUTE(x_1, x_2, x_3)) = [1] + x_3
   POL(PLUS(x_1, x_2)) = 0
   POL(a) = 0
   POL(ack(x_1, x_2)) = [1]
   POL(b) = 0
   POL(c) = 0
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(false) = [1]
   POL(p(x_1)) = [1]
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
S tuples:
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c9_1, c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(0, z0) -> c9(PLUS(z0, s(0))) by 
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
S tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c10_1, c11_2, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) by 
   ACK(s(z0), s(s(z1))) -> c11(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1)))
   ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(s(z1))) -> c11(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1)))
   ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0))
S tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(s(z1))) -> c11(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1)))
   ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2


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

(135) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(s(z1))) -> c11(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) by 
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
S tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(0)) -> c11(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) by 
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
S tuples:
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   ACK_2, PLUS_2, PERMUTE_3

Compound Symbols:   c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1


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

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(s(z0), 0) -> c10(ACK(z0, s(0))) by 
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1


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

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(143) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), p(s(0)), c)) by PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), p(s(s(z1))), c)) by PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), p(s(0)), c)) by PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
   p(s(z0)) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2, p_1

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(153) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   p(s(z0)) -> z0

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c4_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1


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

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PERMUTE(false, s(z0), b) -> c4(PERMUTE(ack(z0, ack(s(z0), z0)), z0, c)) by 
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
   PERMUTE(false, s(s(z1)), b) -> c4(PERMUTE(ack(s(z1), ack(s(z1), ack(s(s(z1)), z1))), s(z1), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1, c4_1


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

(157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c))
   PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1, c4_1


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

(159) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(x1))) -> c11(ACK(0, plus(ack(s(0), x1), s(0))), ACK(s(0), s(x1))) by 
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1))))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1, c4_1


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

(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(s(s(z1)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) by 
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), x1)), s(0))), ACK(s(0), s(s(x1))))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1, c4_1


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

(163) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(s(0))) -> c11(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) by 
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c10_1, c1_1, c4_1


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

(165) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) by 
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0))))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(0)) -> c1(ACK(0, plus(s(0), s(0)))) by 
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(169) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1)))) by 
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0))) by 
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(z1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), z1)), s(0))), ACK(s(0), s(s(z1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(173) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, plus(ack(0, ack(s(0), x1)), s(0))), ACK(s(0), s(s(x1)))) by 
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1))))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(s(x1)))) -> c11(ACK(0, ack(0, plus(ack(s(0), x1), s(0)))), ACK(s(0), s(s(x1)))) by 
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1)))))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(s(s(s(z1))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) by 
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), x1))), s(0))), ACK(s(0), s(s(s(x1)))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), x1)), s(0)))), ACK(s(0), s(s(s(x1)))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0))))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(179) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(z0), s(s(s(0)))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) by 
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(181) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(0))) -> c11(ACK(0, plus(ack(0, s(0)), s(0))), ACK(s(0), s(0))) by 
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) by 
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0)))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(185) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(s(0), s(0)))), ACK(s(0), s(0))) by 
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(187) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(s(0), 0), s(0))), ACK(s(s(0)), 0)) by 
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(189) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(s(z0)), s(0)) -> c11(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) by 
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0)))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(s(0), s(0)) -> c1(ACK(0, plus(0, s(s(0))))) by 
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))
S tuples:
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2


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

(193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1))) by 
   PLUS(s(s(y0)), z1) -> c13(PLUS(s(y0), s(z1)))
   PLUS(s(z0), s(y1)) -> c13(PLUS(z0, s(s(y1))))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))
   PLUS(s(s(y0)), z1) -> c13(PLUS(s(y0), s(z1)))
   PLUS(s(z0), s(y1)) -> c13(PLUS(z0, s(s(y1))))
S tuples:
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))
   PLUS(s(s(y0)), z1) -> c13(PLUS(s(y0), s(z1)))
   PLUS(s(z0), s(y1)) -> c13(PLUS(z0, s(s(y1))))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PLUS_2, PERMUTE_3, ACK_2

Compound Symbols:   c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2, c13_1


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

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

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(0, z0) -> plus(z0, s(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, 0) -> z0
Tuples:
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))
   PLUS(s(s(y0)), z1) -> c13(PLUS(s(y0), s(z1)))
   PLUS(s(z0), s(y1)) -> c13(PLUS(z0, s(s(y1))))
   PLUS(z0, s(s(s(s(y1))))) -> c14(PLUS(s(z0), s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c14(PLUS(s(s(y0)), z1))
   PLUS(z0, s(s(s(y1)))) -> c14(PLUS(s(z0), s(y1)))
S tuples:
   PERMUTE(s(z0), x1, a) -> c2(PERMUTE(false, s(z0), b))
   PERMUTE(false, s(s(z0)), b) -> c3(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c), ACK(s(s(z0)), s(s(z0))))
   PERMUTE(z0, s(y0), c) -> c6(PERMUTE(s(y0), z0, a))
   ACK(0, s(y0)) -> c9(PLUS(s(y0), s(0)))
   ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0))
   ACK(s(0), 0) -> c10(ACK(0, s(0)))
   ACK(s(s(s(y0))), 0) -> c10(ACK(s(s(y0)), s(0)))
   ACK(s(s(0)), 0) -> c10(ACK(s(0), s(0)))
   ACK(s(s(y0)), 0) -> c10(ACK(s(y0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(s(0), 0))
   PERMUTE(false, s(s(z0)), b) -> c4(PERMUTE(ack(s(z0), ack(s(z0), ack(s(s(z0)), z0))), s(z0), c))
   ACK(s(0), s(s(x0))) -> c11(ACK(0, s(ack(s(0), x0))), ACK(s(0), s(x0)))
   ACK(s(0), s(s(x0))) -> c11(ACK(s(0), s(x0)))
   ACK(s(x0), s(s(s(x1)))) -> c11(ACK(s(x0), s(s(x1))))
   ACK(s(x0), s(s(0))) -> c11(ACK(s(x0), s(0)))
   ACK(s(0), s(0)) -> c1(ACK(0, s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, s(ack(0, ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, plus(plus(ack(s(0), x0), s(0)), s(0))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, plus(ack(0, ack(0, ack(s(0), z1))), s(0))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, plus(ack(0, ack(0, s(0))), s(0))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(0))) -> c11(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, plus(plus(s(0), s(0)), s(0))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(s(0), s(0)))
   ACK(s(0), s(s(s(x0)))) -> c11(ACK(0, ack(0, s(ack(s(0), x0)))), ACK(s(0), s(s(x0))))
   ACK(s(0), s(s(s(s(z1))))) -> c11(ACK(0, ack(0, plus(ack(0, ack(s(0), z1)), s(0)))), ACK(s(0), s(s(s(z1)))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, plus(ack(0, s(0)), s(0)))), ACK(s(0), s(s(0))))
   ACK(s(0), s(s(s(s(x1))))) -> c11(ACK(0, ack(0, ack(0, plus(ack(s(0), x1), s(0))))), ACK(s(0), s(s(s(x1)))))
   ACK(s(z0), s(s(s(s(s(z1)))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1))))))
   ACK(s(z0), s(s(s(s(0))))) -> c11(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0)))))
   ACK(s(x0), s(s(s(s(x1))))) -> c11(ACK(s(x0), s(s(s(x1)))))
   ACK(s(s(z0)), s(s(s(0)))) -> c11(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0))))
   ACK(s(0), s(s(s(0)))) -> c11(ACK(0, ack(0, ack(0, plus(s(0), s(0))))), ACK(s(0), s(s(0))))
   ACK(s(x0), s(s(s(0)))) -> c11(ACK(s(x0), s(s(0))))
   ACK(s(s(0)), s(s(0))) -> c11(ACK(s(0), ack(s(0), plus(ack(s(0), 0), s(0)))), ACK(s(s(0)), s(0)))
   ACK(s(s(z0)), s(s(0))) -> c11(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0)))
   ACK(s(s(x0)), s(s(0))) -> c11(ACK(s(s(x0)), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, plus(0, s(s(0))))), ACK(s(0), s(0)))
   ACK(s(0), s(s(0))) -> c11(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0)))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), plus(ack(0, s(0)), s(0))), ACK(s(s(0)), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(s(0)), 0))
   ACK(s(s(s(z0))), s(0)) -> c11(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0))
   ACK(s(s(0)), s(0)) -> c11(ACK(s(0), ack(0, plus(s(0), s(0)))), ACK(s(s(0)), 0))
   ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0))
   ACK(s(0), s(0)) -> c1(ACK(0, s(plus(s(0), 0))))
   PLUS(s(s(y0)), z1) -> c13(PLUS(s(y0), s(z1)))
   PLUS(s(z0), s(y1)) -> c13(PLUS(z0, s(s(y1))))
   PLUS(z0, s(s(s(s(y1))))) -> c14(PLUS(s(z0), s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c14(PLUS(s(s(y0)), z1))
   PLUS(z0, s(s(s(y1)))) -> c14(PLUS(s(z0), s(y1)))
K tuples:
   PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0)))
   PERMUTE(false, s(x0), b) -> c3(ACK(s(x0), s(x0)))
Defined Rule Symbols:   ack_2, plus_2

Defined Pair Symbols:   PERMUTE_3, ACK_2, PLUS_2

Compound Symbols:   c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c10_1, c1_1, c4_1, c11_2, c13_1, c14_1