KILLED proof of input_cRLwpDZ7Y8.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtUsableRulesProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 27 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 35 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) 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: numbers -> d(0) d(x) -> if(le(x, nr), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(0, x) -> s(x) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 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: numbers -> d(0') d(x) -> if(le(x, nr), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) nr -> ack(s(s(s(s(s(s(0')))))), 0') ack(0', x) -> s(x) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 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: numbers -> d(0) d(x) -> if(le(x, nr), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(0, x) -> s(x) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, ack(s(x), y)) 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: numbers -> d(0) [1] d(x) -> if(le(x, nr), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] nr -> ack(s(s(s(s(s(s(0)))))), 0) [1] ack(0, x) -> s(x) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [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: numbers -> d(0) [1] d(x) -> if(le(x, nr), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] nr -> ack(s(s(s(s(s(s(0)))))), 0) [1] ack(0, x) -> s(x) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1] The TRS has the following type information: numbers :: cons:nil d :: 0:s -> cons:nil 0 :: 0:s if :: true:false -> 0:s -> cons:nil le :: 0:s -> 0:s -> true:false nr :: 0:s true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false nil :: cons:nil ack :: 0:s -> 0:s -> 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: numbers d_1 if_2 (c) The following functions are completely defined: ack_2 le_2 nr Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: numbers -> d(0) [1] d(x) -> if(le(x, nr), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] nr -> ack(s(s(s(s(s(s(0)))))), 0) [1] ack(0, x) -> s(x) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1] The TRS has the following type information: numbers :: cons:nil d :: 0:s -> cons:nil 0 :: 0:s if :: true:false -> 0:s -> cons:nil le :: 0:s -> 0:s -> true:false nr :: 0:s true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false nil :: cons:nil ack :: 0:s -> 0:s -> 0:s 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: numbers -> d(0) [1] d(x) -> if(le(x, ack(s(s(s(s(s(s(0)))))), 0)), x) [2] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] nr -> ack(s(s(s(s(s(s(0)))))), 0) [1] ack(0, x) -> s(x) [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] The TRS has the following type information: numbers :: cons:nil d :: 0:s -> cons:nil 0 :: 0:s if :: true:false -> 0:s -> cons:nil le :: 0:s -> 0:s -> true:false nr :: 0:s true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false nil :: cons:nil ack :: 0:s -> 0:s -> 0:s 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: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: 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, 1 + 0) :|: x >= 0, z = 1 + x, z' = 0 ack(z, z') -{ 1 }-> 1 + x :|: z' = x, x >= 0, z = 0 d(z) -{ 2 }-> if(le(x, ack(1 + (1 + (1 + (1 + (1 + (1 + 0))))), 0)), x) :|: x >= 0, z = x if(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 if(z, z') -{ 1 }-> 1 + x + d(1 + x) :|: z' = x, z = 1, x >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 nr -{ 1 }-> ack(1 + (1 + (1 + (1 + (1 + (1 + 0))))), 0) :|: numbers -{ 1 }-> d(0) :|: ---------------------------------------- (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') -{ 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, 1 + 0) :|: z - 1 >= 0, z' = 0 ack(z, z') -{ 1 }-> 1 + z' :|: z' >= 0, z = 0 d(z) -{ 2 }-> if(le(z, ack(1 + (1 + (1 + (1 + (1 + (1 + 0))))), 0)), z) :|: z >= 0 if(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if(z, z') -{ 1 }-> 1 + z' + d(1 + z') :|: z = 1, z' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 nr -{ 1 }-> ack(1 + (1 + (1 + (1 + (1 + (1 + 0))))), 0) :|: numbers -{ 1 }-> d(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: none And the following fresh constants: none ---------------------------------------- (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: numbers -> d(0) [1] d(x) -> if(le(x, nr), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] nr -> ack(s(s(s(s(s(s(0)))))), 0) [1] ack(0, x) -> s(x) [1] ack(s(x), 0) -> ack(x, s(0)) [1] ack(s(x), s(y)) -> ack(x, ack(s(x), y)) [1] The TRS has the following type information: numbers :: cons:nil d :: 0:s -> cons:nil 0 :: 0:s if :: true:false -> 0:s -> cons:nil le :: 0:s -> 0:s -> true:false nr :: 0:s true :: true:false cons :: 0:s -> cons:nil -> cons:nil s :: 0:s -> 0:s false :: true:false nil :: cons:nil ack :: 0:s -> 0:s -> 0:s 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: 0 => 0 true => 1 false => 0 nil => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 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') -{ 1 }-> 1 + x :|: z' = x, x >= 0, z = 0 d(z) -{ 1 }-> if(le(x, nr), x) :|: x >= 0, z = x if(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 if(z, z') -{ 1 }-> 1 + x + d(1 + x) :|: z' = x, z = 1, x >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 nr -{ 1 }-> ack(1 + (1 + (1 + (1 + (1 + (1 + 0))))), 0) :|: numbers -{ 1 }-> d(0) :|: 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: numbers -> d(0) d(z0) -> if(le(z0, nr), z0) if(true, z0) -> cons(z0, d(s(z0))) if(false, z0) -> nil le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: NUMBERS -> c(D(0)) D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) IF(true, z0) -> c2(D(s(z0))) IF(false, z0) -> c3 LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(0, z0) -> c8 ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: NUMBERS -> c(D(0)) D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) IF(true, z0) -> c2(D(s(z0))) IF(false, z0) -> c3 LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(0, z0) -> c8 ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: numbers, d_1, if_2, le_2, nr, ack_2 Defined Pair Symbols: NUMBERS, D_1, IF_2, LE_2, NR, ACK_2 Compound Symbols: c_1, c1_3, c2_1, c3, c4, c5, c6_1, c7_1, c8, c9_1, c10_2 ---------------------------------------- (23) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: NUMBERS -> c(D(0)) Removed 4 trailing nodes: ACK(0, z0) -> c8 IF(false, z0) -> c3 LE(0, z0) -> c4 LE(s(z0), 0) -> c5 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: numbers -> d(0) d(z0) -> if(le(z0, nr), z0) if(true, z0) -> cons(z0, d(s(z0))) if(false, z0) -> nil le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(0, z0) -> s(z0) ack(s(z0), 0) -> ack(z0, s(0)) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: numbers, d_1, if_2, le_2, nr, ack_2 Defined Pair Symbols: D_1, IF_2, LE_2, NR, ACK_2 Compound Symbols: c1_3, c2_1, c6_1, c7_1, c9_1, c10_2 ---------------------------------------- (25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: numbers -> d(0) d(z0) -> if(le(z0, nr), z0) if(true, z0) -> cons(z0, d(s(z0))) if(false, z0) -> nil ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) S tuples: D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: D_1, IF_2, LE_2, NR, ACK_2 Compound Symbols: c1_3, c2_1, c6_1, c7_1, c9_1, c10_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(z0) -> c1(IF(le(z0, nr), z0), LE(z0, nr), NR) by D(0) -> c1(IF(true, 0), LE(0, nr), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(0) -> c1(IF(true, 0), LE(0, nr), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(0) -> c1(IF(true, 0), LE(0, nr), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) D(0) -> c1(IF(true, 0), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) D(0) -> c1(IF(true, 0), NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c1_2 ---------------------------------------- (31) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) D(0) -> c(IF(true, 0)) D(0) -> c(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) D(0) -> c(IF(true, 0)) D(0) -> c(NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c_1 ---------------------------------------- (33) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: D(0) -> c(IF(true, 0)) D(0) -> c(NR) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(z1)) -> c10(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1)) by ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c1_3, c10_2 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(s(0)))))), 0)), x0), LE(x0, nr), NR) by D(0) -> c1(IF(true, 0), LE(0, nr), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c1(LE(x0, nr), NR) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(0) -> c1(IF(true, 0), LE(0, nr), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c1(LE(x0, nr), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(0) -> c1(IF(true, 0), LE(0, nr), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c1(LE(x0, nr), NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c1_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c1(LE(x0, nr), NR) D(0) -> c1(IF(true, 0), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c1(LE(x0, nr), NR) D(0) -> c1(IF(true, 0), NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c1_2 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) D(0) -> c(IF(true, 0)) D(0) -> c(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) D(0) -> c(IF(true, 0)) D(0) -> c(NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: D(0) -> c(IF(true, 0)) D(0) -> c(NR) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) K tuples:none Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c_1 ---------------------------------------- (45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) We considered the (Usable) Rules:none And the Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ACK(x_1, x_2)) = 0 POL(D(x_1)) = [1] + x_1 POL(IF(x_1, x_2)) = [1] + x_2 POL(LE(x_1, x_2)) = 0 POL(NR) = 0 POL(ack(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(nr) = [1] POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(0)) -> c10(ACK(z0, ack(z0, s(0))), ACK(s(z0), 0)) by ACK(s(0), s(0)) -> c10(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) ACK(s(0), s(0)) -> c10(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) ACK(s(0), s(0)) -> c10(ACK(0, s(s(0))), ACK(s(0), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c_1, c10_1 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c10_2, c1_3, c_1, c10_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(z1))) -> c10(ACK(z0, ack(z0, ack(s(z0), z1))), ACK(s(z0), s(z1))) by ACK(s(0), s(s(x1))) -> c10(ACK(0, s(ack(s(0), x1))), ACK(s(0), s(x1))) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(0, s(ack(s(0), x1))), ACK(s(0), s(x1))) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(0, s(ack(s(0), x1))), ACK(s(0), s(x1))) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c1_3, c_1, c10_2, c10_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c1_3, c_1, c10_2, c10_1 ---------------------------------------- (55) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(x0) -> c1(IF(le(x0, ack(s(s(s(s(s(0))))), s(0))), x0), LE(x0, nr), NR) by D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_2, c10_1, c1_3 ---------------------------------------- (57) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(x0) -> c(LE(x0, nr)) by D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) nr -> ack(s(s(s(s(s(s(0)))))), 0) ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: le_2, nr, ack_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_2, c10_1, c1_3 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: nr -> ack(s(s(s(s(s(s(0)))))), 0) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_2, c10_1, c1_3 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(s(z0), 0))), ACK(s(s(z0)), 0)) by ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(0))) -> c10(ACK(z0, ack(z0, ack(z0, s(0)))), ACK(s(z0), s(0))) by ACK(s(0), s(s(0))) -> c10(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(0), s(s(0))) -> c10(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(0), s(s(0))) -> c10(ACK(0, s(ack(0, s(0)))), ACK(s(0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(0, ack(0, s(s(0)))), ACK(s(0), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(s(z1)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(s(z0), z1)))), ACK(s(z0), s(s(z1)))) by ACK(s(0), s(s(s(x1)))) -> c10(ACK(0, s(ack(0, ack(s(0), x1)))), ACK(s(0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(0, ack(0, s(ack(s(0), x1)))), ACK(s(0), s(s(x1)))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(0, s(ack(0, ack(s(0), x1)))), ACK(s(0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(0, ack(0, s(ack(s(0), x1)))), ACK(s(0), s(s(x1)))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(0, s(ack(0, ack(s(0), x1)))), ACK(s(0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(0, ack(0, s(ack(s(0), x1)))), ACK(s(0), s(s(x1)))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c1_3, c10_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c1_3, c10_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(z0) -> c1(IF(le(z0, ack(s(s(s(s(s(0))))), s(0))), z0), LE(z0, ack(s(s(s(s(s(s(0)))))), 0)), NR) by D(0) -> c1(IF(true, 0), LE(0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(0) -> c1(IF(true, 0), LE(0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(0) -> c1(IF(true, 0), LE(0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c1_2 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(0) -> c1(IF(true, 0), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(0) -> c1(IF(true, 0), NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c1_2 ---------------------------------------- (75) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(0) -> c3(IF(true, 0)) D(0) -> c3(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(0) -> c3(IF(true, 0)) D(0) -> c3(NR) K tuples: D(x0) -> c(LE(x0, nr)) D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (77) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: D(0) -> c3(IF(true, 0)) D(0) -> c3(NR) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) K tuples: D(x0) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) We considered the (Usable) Rules: le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ACK(x_1, x_2)) = 0 POL(D(x_1)) = [1] + x_1 POL(IF(x_1, x_2)) = x_1 POL(LE(x_1, x_2)) = 0 POL(NR) = 0 POL(ack(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(z0) -> c(LE(z0, ack(s(s(s(s(s(s(0)))))), 0))) by D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(s(0), 0))), ACK(s(s(0)), 0)) by ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(s(z0)), s(0)) -> c10(ACK(s(z0), ack(z0, ack(z0, s(0)))), ACK(s(s(z0)), 0)) by ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(z0)), s(0))) by ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(s(0)))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, s(0))))), ACK(s(z0), s(s(0)))) by ACK(s(0), s(s(s(0)))) -> c10(ACK(0, s(ack(0, ack(0, s(0))))), ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, ack(0, s(ack(0, s(0))))), ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, ack(0, ack(0, s(s(0))))), ACK(s(0), s(s(0)))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, s(ack(0, ack(0, s(0))))), ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, ack(0, s(ack(0, s(0))))), ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, ack(0, ack(0, s(s(0))))), ACK(s(0), s(s(0)))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, s(ack(0, ack(0, s(0))))), ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, ack(0, s(ack(0, s(0))))), ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(0, ack(0, ack(0, s(s(0))))), ACK(s(0), s(s(0)))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c10_2, c1_3, c3_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ACK(s(z0), s(s(s(s(z1))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1))))), ACK(s(z0), s(s(s(z1))))) by ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, s(ack(0, ack(0, ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, ack(0, s(ack(0, ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, ack(0, ack(0, s(ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, s(ack(0, ack(0, ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, ack(0, s(ack(0, ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, ack(0, ack(0, s(ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, s(ack(0, ack(0, ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, ack(0, s(ack(0, ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(0, ack(0, ack(0, s(ack(s(0), x1))))), ACK(s(0), s(s(s(x1))))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c1_3, c3_1, c10_2 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c1_3, c3_1, c10_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) by D(0) -> c1(IF(true, 0), LE(0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(0) -> c1(IF(true, 0), LE(0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(0) -> c1(IF(true, 0), LE(0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c1_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(0) -> c1(IF(true, 0), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c1(LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(0) -> c1(IF(true, 0), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c1_2 ---------------------------------------- (101) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(0) -> c4(IF(true, 0)) D(0) -> c4(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(0) -> c4(IF(true, 0)) D(0) -> c4(NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1 ---------------------------------------- (103) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: D(0) -> c4(IF(true, 0)) D(0) -> c4(NR) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) We considered the (Usable) Rules: le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ACK(x_1, x_2)) = 0 POL(D(x_1)) = [1] + x_1 POL(IF(x_1, x_2)) = x_1 + x_2 POL(LE(x_1, x_2)) = 0 POL(NR) = 0 POL(ack(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) by D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace D(x0) -> c(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) by D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) LE(s(z0), s(z1)) -> c6(LE(z0, z1)) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, LE_2, NR, ACK_2, D_1 Compound Symbols: c2_1, c6_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c6(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1, c6_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) by ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1, c6_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(ack(0, s(0)))), ACK(s(s(0)), 0)) by ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1, c6_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(0)), s(0)) -> c10(ACK(s(0), ack(0, s(s(0)))), ACK(s(s(0)), 0)) by ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1, c6_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(s(z0), 0)))), ACK(s(s(s(z0))), 0)) by ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1, c6_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(s(0), s(ack(s(0), 0)))), ACK(s(s(0)), s(0))) by ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) D(x0) -> c(NR) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) K tuples: D(x0) -> c(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c_1, c10_1, c3_1, c10_2, c1_3, c4_1, c6_1 ---------------------------------------- (123) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(x0) -> c(NR) by D(s(x0)) -> c(NR) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) D(x0) -> c3(NR) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) K tuples: D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c3(NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c10_1, c3_1, c10_2, c1_3, c4_1, c_1, c6_1 ---------------------------------------- (125) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(x0) -> c3(NR) by D(s(x0)) -> c3(NR) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) K tuples: D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c10_1, c10_2, c1_3, c4_1, c3_1, c_1, c6_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(s(z0), 0))))), ACK(s(s(z0)), s(s(0)))) by ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(z0), 0) -> c9(ACK(z0, s(0))) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) K tuples: D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c9_1, c10_1, c10_2, c1_3, c4_1, c3_1, c_1, c6_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(z0), 0) -> c9(ACK(z0, s(0))) by ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) K tuples: D(x0) -> c3(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c1_3, c4_1, c3_1, c_1, c6_1, c9_1 ---------------------------------------- (131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ACK(s(0), s(0)) -> c10(ACK(s(0), 0)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c1_3, c4_1, c3_1, c_1, c6_1, c9_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(x0) -> c1(IF(le(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), x0), LE(x0, ack(s(s(s(s(s(s(0)))))), 0)), NR) by D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c4_1, c3_1, c_1, c6_1, c9_1, c1_3 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) by D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c4(NR) D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c4_1, c3_1, c_1, c6_1, c9_1, c1_3 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(x0) -> c3(LE(x0, ack(s(s(s(s(s(0))))), s(0)))) by D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c4(NR) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(x0) -> c4(NR) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c4_1, c_1, c6_1, c3_1, c9_1, c1_3 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(x0) -> c4(NR) by D(s(x0)) -> c4(NR) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, D_1, LE_2 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c_1, c6_1, c3_1, c9_1, c1_3, c4_1 ---------------------------------------- (141) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(x0) -> c(LE(x0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) by D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c1_3, c4_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), ack(0, ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) by ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c1_3, c4_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(x0), s(0)) -> c10(ACK(s(x0), 0)) by ACK(s(s(y0)), s(0)) -> c10(ACK(s(s(y0)), 0)) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) K tuples: D(x0) -> c4(LE(x0, ack(s(s(s(s(s(s(0)))))), 0))) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c1_3, c4_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ACK(s(s(0)), 0) -> c9(ACK(s(0), s(0))) ACK(s(0), s(s(0))) -> c10(ACK(s(0), s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c1_3, c4_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(s(0))))), s(0))), NR) by D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(z0) -> c4(LE(z0, ack(s(s(s(s(s(0))))), s(0)))) by D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3 ---------------------------------------- (153) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(z0) -> c3(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) by D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACK(s(0), s(s(x1))) -> c10(ACK(s(0), s(x1))) by ACK(s(0), s(s(s(y0)))) -> c10(ACK(s(0), s(s(y0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(0), s(s(s(s(y1))))) -> c10(ACK(s(0), s(s(s(y1))))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3 ---------------------------------------- (157) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(z0) -> c1(IF(le(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), z0), LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) by D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) S tuples: IF(true, z0) -> c2(D(s(z0))) NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: IF_2, NR, ACK_2, LE_2, D_1 Compound Symbols: c2_1, c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3 ---------------------------------------- (159) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, z0) -> c2(D(s(z0))) by IF(true, s(x0)) -> c2(D(s(s(x0)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) S tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: NR, ACK_2, LE_2, D_1, IF_2 Compound Symbols: c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3, c2_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace D(s(x0)) -> c(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) by D(s(z0)) -> c(LE(s(z0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) D(s(z0)) -> c(LE(s(z0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0))))) S tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) K tuples: D(s(x0)) -> c(NR) D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: NR, ACK_2, LE_2, D_1, IF_2 Compound Symbols: c7_1, c10_1, c10_2, c6_1, c_1, c3_1, c9_1, c4_1, c1_3, c2_1 ---------------------------------------- (163) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(s(x0)) -> c(NR) by D(s(s(x0))) -> c(NR) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) D(s(x0)) -> c3(NR) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) D(s(z0)) -> c(LE(s(z0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0))))) D(s(s(x0))) -> c(NR) S tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) K tuples: D(s(x0)) -> c3(NR) D(s(x0)) -> c4(NR) D(s(s(x0))) -> c(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: NR, ACK_2, LE_2, D_1, IF_2 Compound Symbols: c7_1, c10_1, c10_2, c6_1, c3_1, c9_1, c4_1, c1_3, c2_1, c_1 ---------------------------------------- (165) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(s(x0)) -> c3(NR) by D(s(s(x0))) -> c3(NR) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) D(s(x0)) -> c4(NR) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) D(s(z0)) -> c(LE(s(z0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0))))) D(s(s(x0))) -> c(NR) D(s(s(x0))) -> c3(NR) S tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) K tuples: D(s(x0)) -> c4(NR) D(s(s(x0))) -> c(NR) D(s(s(x0))) -> c3(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: NR, ACK_2, LE_2, D_1, IF_2 Compound Symbols: c7_1, c10_1, c10_2, c6_1, c9_1, c4_1, c3_1, c1_3, c2_1, c_1 ---------------------------------------- (167) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(s(x0)) -> c4(NR) by D(s(s(x0))) -> c4(NR) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) D(s(z0)) -> c(LE(s(z0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0))))) D(s(s(x0))) -> c(NR) D(s(s(x0))) -> c3(NR) D(s(s(x0))) -> c4(NR) S tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) K tuples: D(s(s(x0))) -> c(NR) D(s(s(x0))) -> c3(NR) D(s(s(x0))) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: NR, ACK_2, LE_2, D_1, IF_2 Compound Symbols: c7_1, c10_1, c10_2, c6_1, c9_1, c4_1, c3_1, c1_3, c2_1, c_1 ---------------------------------------- (169) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace D(z0) -> c4(LE(z0, ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) by D(s(s(x0))) -> c4(LE(s(s(x0)), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: ack(s(z0), 0) -> ack(z0, s(0)) ack(0, z0) -> s(z0) ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) Tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) D(s(x0)) -> c3(LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) D(s(z0)) -> c(LE(s(z0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0))))) D(s(s(x0))) -> c(NR) D(s(s(x0))) -> c3(NR) D(s(s(x0))) -> c4(NR) D(s(s(x0))) -> c4(LE(s(s(x0)), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0)))) S tuples: NR -> c7(ACK(s(s(s(s(s(s(0)))))), 0)) ACK(s(s(x0)), s(0)) -> c10(ACK(s(s(x0)), 0)) ACK(s(x0), s(s(0))) -> c10(ACK(s(x0), s(0))) ACK(s(x0), s(s(s(x1)))) -> c10(ACK(s(x0), s(s(x1)))) ACK(s(0), s(s(s(x1)))) -> c10(ACK(s(0), s(s(x1)))) ACK(s(s(0)), s(0)) -> c10(ACK(s(s(0)), 0)) ACK(s(s(z0)), s(s(0))) -> c10(ACK(s(z0), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(z0)), s(0))) ACK(s(s(x0)), s(s(0))) -> c10(ACK(s(s(x0)), s(0))) ACK(s(x0), s(s(s(0)))) -> c10(ACK(s(x0), s(s(0)))) ACK(s(0), s(s(s(0)))) -> c10(ACK(s(0), s(s(0)))) ACK(s(z0), s(s(s(s(0))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, s(0)))))), ACK(s(z0), s(s(s(0))))) ACK(s(z0), s(s(s(s(s(z1)))))) -> c10(ACK(z0, ack(z0, ack(z0, ack(z0, ack(z0, ack(s(z0), z1)))))), ACK(s(z0), s(s(s(s(z1)))))) ACK(s(x0), s(s(s(s(x1))))) -> c10(ACK(s(x0), s(s(s(x1))))) ACK(s(0), s(s(s(s(x1))))) -> c10(ACK(s(0), s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c6(LE(s(y0), s(y1))) ACK(s(s(0)), s(0)) -> c10(ACK(s(0), s(s(s(0)))), ACK(s(s(0)), 0)) ACK(s(s(s(z0))), s(0)) -> c10(ACK(s(s(z0)), ack(s(z0), ack(z0, ack(z0, s(0))))), ACK(s(s(s(z0))), 0)) ACK(s(s(z0)), s(s(s(0)))) -> c10(ACK(s(z0), ack(s(z0), ack(s(z0), ack(z0, ack(z0, s(0)))))), ACK(s(s(z0)), s(s(0)))) ACK(s(s(y0)), 0) -> c9(ACK(s(y0), s(0))) ACK(s(s(s(y0))), 0) -> c9(ACK(s(s(y0)), s(0))) ACK(s(s(s(0))), 0) -> c9(ACK(s(s(0)), s(0))) ACK(s(s(s(s(y0)))), 0) -> c9(ACK(s(s(s(y0))), s(0))) ACK(s(s(0)), s(s(0))) -> c10(ACK(s(0), s(ack(s(0), ack(s(0), 0)))), ACK(s(s(0)), s(0))) ACK(s(s(s(y0))), s(0)) -> c10(ACK(s(s(s(y0))), 0)) ACK(s(s(s(0))), s(0)) -> c10(ACK(s(s(s(0))), 0)) ACK(s(s(s(s(y0)))), s(0)) -> c10(ACK(s(s(s(s(y0)))), 0)) ACK(s(0), s(s(s(s(0))))) -> c10(ACK(s(0), s(s(s(0))))) ACK(s(0), s(s(s(s(s(0)))))) -> c10(ACK(s(0), s(s(s(s(0)))))) ACK(s(0), s(s(s(s(s(s(y1))))))) -> c10(ACK(s(0), s(s(s(s(s(y1))))))) ACK(s(0), s(s(s(s(s(y1)))))) -> c10(ACK(s(0), s(s(s(s(y1)))))) D(s(x0)) -> c1(IF(le(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(0)))), s(0)))), s(x0)), LE(s(x0), ack(s(s(s(s(0)))), ack(s(s(s(s(s(0))))), 0))), NR) IF(true, s(x0)) -> c2(D(s(s(x0)))) K tuples: D(s(s(x0))) -> c(NR) D(s(s(x0))) -> c3(NR) D(s(s(x0))) -> c4(NR) Defined Rule Symbols: ack_2, le_2 Defined Pair Symbols: NR, ACK_2, LE_2, D_1, IF_2 Compound Symbols: c7_1, c10_1, c10_2, c6_1, c9_1, c3_1, c1_3, c2_1, c_1, c4_1