KILLED proof of input_s23OXtuTti.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 52 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 23 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtLeafRemovalProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms] (178) 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) 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 ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 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 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (13) 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 ---------------------------------------- (14) 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 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: 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 ---------------------------------------- (17) 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 ---------------------------------------- (18) 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 ---------------------------------------- (19) 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 ---------------------------------------- (20) 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. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) 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 ---------------------------------------- (23) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: DOUBLE(z0) -> c1(PERMUTE(z0, z0, a)) Removed 8 trailing nodes: PLUS(0, z0) -> c12 ISZERO(0) -> c17 P(s(z0)) -> c8 P(0) -> c7 PERMUTE(true, z0, b) -> c5 PLUS(z0, 0) -> c16 ISZERO(s(z0)) -> c18 PLUS(z0, s(0)) -> c15 ---------------------------------------- (24) 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 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (26) 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 ---------------------------------------- (27) 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))) ---------------------------------------- (28) 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 ---------------------------------------- (29) 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)) ---------------------------------------- (30) 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 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PERMUTE(0, x1, a) -> c2(PERMUTE(true, 0, b)) ---------------------------------------- (32) 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 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: isZero(0) -> true isZero(s(z0)) -> false ---------------------------------------- (34) 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 ---------------------------------------- (35) 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))) ---------------------------------------- (36) 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 ---------------------------------------- (37) 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))) ---------------------------------------- (38) 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 ---------------------------------------- (39) 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))) ---------------------------------------- (40) 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 ---------------------------------------- (41) 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))) ---------------------------------------- (42) 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 ---------------------------------------- (43) 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)) ---------------------------------------- (44) 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)), 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))) 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)), 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))) 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 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: p(0) -> 0 ---------------------------------------- (46) 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)), 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))) 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)), 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))) 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 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing 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))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) ---------------------------------------- (48) 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(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(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c)) 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(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(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(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(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c)) 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(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(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, c2_1, c3_2, c4_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) ---------------------------------------- (50) 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(ack(0, ack(0, s(0))), p(s(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(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), p(s(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(plus(ack(s(0), 0), s(0)), 0, c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing 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))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(0)), c), ACK(s(0), s(0))) ---------------------------------------- (52) 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(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c)) 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(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(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(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(x0), b) -> c4(PERMUTE(ack(s(x0), s(x0)), p(s(x0)), c)) 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(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(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, c2_1, c3_2, c4_1 ---------------------------------------- (53) 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)) ---------------------------------------- (54) 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(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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)) 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(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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: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 ---------------------------------------- (55) 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))) ---------------------------------------- (56) 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(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(s(0), 0)), p(s(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) ---------------------------------------- (58) 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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(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(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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(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, c2_1, c3_2, c4_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) ---------------------------------------- (60) 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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(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, c2_1, c3_2, c4_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(s(s(z1))), b) -> c3(PERMUTE(ack(s(s(z1)), ack(s(s(z1)), ack(s(s(z1)), ack(s(s(s(z1))), z1)))), p(s(s(s(z1)))), c), ACK(s(s(s(z1))), s(s(s(z1))))) by PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) ---------------------------------------- (62) 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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(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 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) by PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) ---------------------------------------- (64) 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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, ack(0, s(0))), 0, c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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 ---------------------------------------- (65) 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))) ---------------------------------------- (66) 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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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 ---------------------------------------- (67) 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(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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] POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] ---------------------------------------- (68) 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(0, s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(ack(0, s(0)), s(0)), p(s(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) ---------------------------------------- (70) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(s(0), s(0))), p(s(0)), c), ACK(s(0), s(0))) by 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(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(0)), c), ACK(s(0), s(0))) ---------------------------------------- (72) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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 ---------------------------------------- (73) 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)) ---------------------------------------- (74) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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, c4_1, c3_1 ---------------------------------------- (75) 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)) ---------------------------------------- (76) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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 ---------------------------------------- (77) 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)) ---------------------------------------- (78) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) 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(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(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(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing tuple parts ---------------------------------------- (80) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (81) 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))) ---------------------------------------- (82) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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 ---------------------------------------- (83) 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))) ---------------------------------------- (84) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (86) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (87) 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))) ---------------------------------------- (88) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (89) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (90) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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))) 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 ---------------------------------------- (91) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (93) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (95) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (97) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) 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))) 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 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(0), b) -> c3(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, 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 ---------------------------------------- (103) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), 0, c), ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) 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))) 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 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(0), b) -> c3(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, 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 ---------------------------------------- (113) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(s(s(s(z1)))), b) -> c3(PERMUTE(ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(z1))), ack(s(s(s(s(z1)))), z1))))), p(s(s(s(s(z1))))), c), ACK(s(s(s(s(z1)))), s(s(s(s(z1)))))) by PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) 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 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), p(s(s(s(0)))), c), ACK(s(s(s(0))), s(s(s(0))))) by PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(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, 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 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), p(s(s(0))), c), ACK(s(s(0)), s(s(0)))) by PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), p(s(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))) ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(s(ack(0, s(0))), 0, c), ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(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))) ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(plus(plus(s(0), s(0)), s(0)), p(s(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))) ---------------------------------------- (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(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), p(s(0)), c), ACK(s(0), s(0))) by PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, plus(0, s(s(0)))), 0, c), 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, 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 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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))) 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 ---------------------------------------- (137) 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))) 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))) ---------------------------------------- (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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(ACK(s(0), s(0))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), p(s(0)), c), ACK(s(0), s(0))) by PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), 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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c3(PERMUTE(ack(0, s(s(0))), 0, c), ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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))) 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 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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))) 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 ---------------------------------------- (143) 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))) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (145) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c)) 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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 ---------------------------------------- (147) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(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, 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 ---------------------------------------- (149) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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 ---------------------------------------- (151) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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 ---------------------------------------- (153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c)) ---------------------------------------- (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 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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 ---------------------------------------- (155) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: p(s(z0)) -> z0 ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(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)) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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: 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 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(s(0), s(0)))), s(0), c), ACK(s(s(0)), s(s(0)))) by PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(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: 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 ---------------------------------------- (159) 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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) We considered the (Usable) Rules:none And the 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(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)) = 0 POL(b) = 0 POL(c) = 0 POL(c10(x_1)) = x_1 POL(c11(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(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 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 ---------------------------------------- (161) 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)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c4_1 ---------------------------------------- (163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: PERMUTE(false, s(0), b) -> c4(PERMUTE(ack(0, ack(0, s(0))), 0, c)) PERMUTE(false, s(0), b) -> c4(PERMUTE(plus(ack(s(0), 0), s(0)), 0, c)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c4_1 ---------------------------------------- (165) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c4_1 ---------------------------------------- (167) 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)))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c4_1 ---------------------------------------- (169) 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))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c4_1 ---------------------------------------- (171) 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)) ACK(s(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0)) ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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(s(x0)), s(0)) -> c11(ACK(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0)) ACK(s(x0), s(0)) -> c11(ACK(s(x0), 0)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_2, c11_1, c4_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(0), s(0)) -> c11(ACK(0, plus(s(0), s(0))), ACK(s(0), 0)) by ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 0)) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c4_1, c11_2 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), s(0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) by PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(0), ack(s(s(0)), 0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(s(s(0))))) ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 0)) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(0), ack(s(s(0)), 0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 0)) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(0), ack(s(s(0)), 0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c4_1, c11_2 ---------------------------------------- (177) 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(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(s(s(0))))) We considered the (Usable) Rules:none And the 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 0)) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(0), ack(s(s(0)), 0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(s(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)) = 0 POL(b) = 0 POL(c) = 0 POL(c10(x_1)) = x_1 POL(c11(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(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 ---------------------------------------- (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: 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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) 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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 0)) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(0), ack(s(s(0)), 0))))), s(s(0)), c), ACK(s(s(s(0))), s(s(s(0))))) PERMUTE(false, s(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(s(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(x0)), b) -> c3(PERMUTE(ack(s(x0), ack(s(x0), ack(s(s(x0)), x0))), s(x0), c), ACK(s(s(x0)), s(s(x0)))) PERMUTE(false, s(s(s(x0))), b) -> c3(PERMUTE(ack(s(s(x0)), ack(s(s(x0)), ack(s(s(x0)), ack(s(s(s(x0))), x0)))), s(s(x0)), c), ACK(s(s(s(x0))), s(s(s(x0))))) 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)) PERMUTE(false, s(s(s(s(z0)))), b) -> c3(PERMUTE(ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(z0))), ack(s(s(s(s(z0)))), z0))))), s(s(s(z0))), c), ACK(s(s(s(s(z0)))), s(s(s(s(z0)))))) PERMUTE(false, s(s(0)), b) -> c3(PERMUTE(ack(s(0), ack(s(0), ack(0, ack(s(0), 0)))), s(0), c), ACK(s(s(0)), s(s(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(s(x0)), s(0)) -> c11(ACK(s(s(x0)), 0)) ACK(s(0), s(0)) -> c11(ACK(0, plus(0, s(s(0)))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(0), s(0)) -> c11(ACK(s(0), 0)) PERMUTE(false, s(s(s(0))), b) -> c3(PERMUTE(ack(s(s(0)), ack(s(s(0)), ack(s(s(0)), ack(s(0), ack(s(s(0)), 0))))), s(s(0)), c), ACK(s(s(s(0))), s(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))) PERMUTE(false, s(s(0)), b) -> c3(ACK(s(s(0)), s(s(0)))) PERMUTE(false, s(s(s(0))), b) -> c3(ACK(s(s(s(0))), s(s(s(0))))) Defined Rule Symbols: ack_2, plus_2 Defined Pair Symbols: ACK_2, PLUS_2, PERMUTE_3 Compound Symbols: c10_1, c13_1, c14_1, c2_1, c3_2, c3_1, c6_1, c9_1, c11_1, c4_1, c11_2