KILLED proof of input_NzJjuMrleB.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 102 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 11 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 1373 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 269 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [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) CdtForwardInstantiationProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) 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: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) fac(0, x) -> x fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) factorial(x) -> fac(x, s(0)) 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: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) fac(0', x) -> x fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) factorial(x) -> fac(x, s(0')) 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: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) fac(0, x) -> x fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) factorial(x) -> fac(x, s(0)) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] fac(0, x) -> x [1] fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) [1] factorial(x) -> fac(x, s(0)) [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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] fac(0, x) -> x [1] fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) [1] factorial(x) -> fac(x, s(0)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s fac :: 0:s -> 0:s -> 0:s factorial :: 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: fac_2 factorial_1 (c) The following functions are completely defined: p_1 times_2 plus_2 Due to the following rules being added: p(v0) -> 0 [0] 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] fac(0, x) -> x [1] fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) [1] factorial(x) -> fac(x, s(0)) [1] p(v0) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s fac :: 0:s -> 0:s -> 0:s factorial :: 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: plus(0, x) -> x [1] plus(s(0), y) -> s(plus(0, y)) [2] plus(s(s(x')), y) -> s(plus(s(p(s(x'))), y)) [2] plus(s(x), y) -> s(plus(0, y)) [1] times(0, y) -> 0 [1] times(s(0), y) -> plus(y, times(0, y)) [2] times(s(s(x'')), y) -> plus(y, times(s(p(s(x''))), y)) [2] times(s(x), y) -> plus(y, times(0, y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] fac(0, x) -> x [1] fac(s(0), y) -> fac(0, plus(y, times(p(s(0)), y))) [3] fac(s(s(x1)), y) -> fac(s(p(s(x1))), plus(y, times(p(s(s(x1))), y))) [3] fac(s(x), y) -> fac(0, plus(y, times(p(s(x)), y))) [2] factorial(x) -> fac(x, s(0)) [1] p(v0) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s fac :: 0:s -> 0:s -> 0:s factorial :: 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 fac(z, z') -{ 2 }-> fac(0, plus(y, times(p(1 + x), y))) :|: x >= 0, y >= 0, z = 1 + x, z' = y fac(z, z') -{ 3 }-> fac(0, plus(y, times(p(1 + 0), y))) :|: z = 1 + 0, y >= 0, z' = y fac(z, z') -{ 3 }-> fac(1 + p(1 + x1), plus(y, times(p(1 + (1 + x1)), y))) :|: z = 1 + (1 + x1), x1 >= 0, y >= 0, z' = y factorial(z) -{ 1 }-> fac(x, 1 + 0) :|: x >= 0, z = x p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, y) :|: z = 1 + 0, y >= 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(0, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y plus(z, z') -{ 2 }-> 1 + plus(1 + p(1 + x'), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y times(z, z') -{ 2 }-> plus(y, times(0, y)) :|: z = 1 + 0, y >= 0, z' = y times(z, z') -{ 1 }-> plus(y, times(0, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 2 }-> plus(y, times(1 + p(1 + x''), y)) :|: y >= 0, x'' >= 0, z = 1 + (1 + x''), z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y ---------------------------------------- (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: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 3 }-> fac(0, plus(z', times(p(1 + 0), z'))) :|: z = 1 + 0, z' >= 0 fac(z, z') -{ 2 }-> fac(0, plus(z', times(p(1 + (z - 1)), z'))) :|: z - 1 >= 0, z' >= 0 fac(z, z') -{ 3 }-> fac(1 + p(1 + (z - 2)), plus(z', times(p(1 + (1 + (z - 2))), z'))) :|: z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 }-> 1 + plus(1 + p(1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(1 + p(1 + (z - 2)), z')) :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { p } { plus } { times } { fac } { factorial } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 3 }-> fac(0, plus(z', times(p(1 + 0), z'))) :|: z = 1 + 0, z' >= 0 fac(z, z') -{ 2 }-> fac(0, plus(z', times(p(1 + (z - 1)), z'))) :|: z - 1 >= 0, z' >= 0 fac(z, z') -{ 3 }-> fac(1 + p(1 + (z - 2)), plus(z', times(p(1 + (1 + (z - 2))), z'))) :|: z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 }-> 1 + plus(1 + p(1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(1 + p(1 + (z - 2)), z')) :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {p}, {plus}, {times}, {fac}, {factorial} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 3 }-> fac(0, plus(z', times(p(1 + 0), z'))) :|: z = 1 + 0, z' >= 0 fac(z, z') -{ 2 }-> fac(0, plus(z', times(p(1 + (z - 1)), z'))) :|: z - 1 >= 0, z' >= 0 fac(z, z') -{ 3 }-> fac(1 + p(1 + (z - 2)), plus(z', times(p(1 + (1 + (z - 2))), z'))) :|: z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 }-> 1 + plus(1 + p(1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(1 + p(1 + (z - 2)), z')) :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {p}, {plus}, {times}, {fac}, {factorial} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 3 }-> fac(0, plus(z', times(p(1 + 0), z'))) :|: z = 1 + 0, z' >= 0 fac(z, z') -{ 2 }-> fac(0, plus(z', times(p(1 + (z - 1)), z'))) :|: z - 1 >= 0, z' >= 0 fac(z, z') -{ 3 }-> fac(1 + p(1 + (z - 2)), plus(z', times(p(1 + (1 + (z - 2))), z'))) :|: z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 }-> 1 + plus(1 + p(1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(1 + p(1 + (z - 2)), z')) :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {p}, {plus}, {times}, {fac}, {factorial} Previous analysis results are: p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 3 }-> fac(0, plus(z', times(p(1 + 0), z'))) :|: z = 1 + 0, z' >= 0 fac(z, z') -{ 2 }-> fac(0, plus(z', times(p(1 + (z - 1)), z'))) :|: z - 1 >= 0, z' >= 0 fac(z, z') -{ 3 }-> fac(1 + p(1 + (z - 2)), plus(z', times(p(1 + (1 + (z - 2))), z'))) :|: z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 }-> 1 + plus(1 + p(1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(1 + p(1 + (z - 2)), z')) :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {fac}, {factorial} Previous analysis results are: p: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 5 }-> fac(0, plus(z', times(s1, z'))) :|: s1 >= 0, s1 <= 1 + 0, z = 1 + 0, z' >= 0 fac(z, z') -{ 3 + z }-> fac(0, plus(z', times(s4, z'))) :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 fac(z, z') -{ 4 + 2*z }-> fac(1 + s2, plus(z', times(s3, z'))) :|: s2 >= 0, s2 <= 1 + (z - 2), s3 >= 0, s3 <= 1 + (1 + (z - 2)), z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 + z }-> 1 + plus(1 + s, z') :|: s >= 0, s <= 1 + (z - 2), z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 + z }-> plus(z', times(1 + s', z')) :|: s' >= 0, s' <= 1 + (z - 2), z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {fac}, {factorial} Previous analysis results are: p: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 5 }-> fac(0, plus(z', times(s1, z'))) :|: s1 >= 0, s1 <= 1 + 0, z = 1 + 0, z' >= 0 fac(z, z') -{ 3 + z }-> fac(0, plus(z', times(s4, z'))) :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 fac(z, z') -{ 4 + 2*z }-> fac(1 + s2, plus(z', times(s3, z'))) :|: s2 >= 0, s2 <= 1 + (z - 2), s3 >= 0, s3 <= 1 + (1 + (z - 2)), z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 + z }-> 1 + plus(1 + s, z') :|: s >= 0, s <= 1 + (z - 2), z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 + z }-> plus(z', times(1 + s', z')) :|: s' >= 0, s' <= 1 + (z - 2), z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {fac}, {factorial} Previous analysis results are: p: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: ?, size: INF ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 fac(z, z') -{ 5 }-> fac(0, plus(z', times(s1, z'))) :|: s1 >= 0, s1 <= 1 + 0, z = 1 + 0, z' >= 0 fac(z, z') -{ 3 + z }-> fac(0, plus(z', times(s4, z'))) :|: s4 >= 0, s4 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 fac(z, z') -{ 4 + 2*z }-> fac(1 + s2, plus(z', times(s3, z'))) :|: s2 >= 0, s2 <= 1 + (z - 2), s3 >= 0, s3 <= 1 + (1 + (z - 2)), z - 2 >= 0, z' >= 0 factorial(z) -{ 1 }-> fac(z, 1 + 0) :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 }-> 1 + plus(0, z') :|: z = 1 + 0, z' >= 0 plus(z, z') -{ 1 }-> 1 + plus(0, z') :|: z - 1 >= 0, z' >= 0 plus(z, z') -{ 2 + z }-> 1 + plus(1 + s, z') :|: s >= 0, s <= 1 + (z - 2), z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', times(0, z')) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> plus(z', times(0, z')) :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 + z }-> plus(z', times(1 + s', z')) :|: s' >= 0, s' <= 1 + (z - 2), z' >= 0, z - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {fac}, {factorial} Previous analysis results are: p: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: INF, size: INF ---------------------------------------- (31) 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: p(v0) -> null_p [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] fac(v0, v1) -> null_fac [0] And the following fresh constants: null_p, null_plus, null_times, null_fac ---------------------------------------- (32) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] fac(0, x) -> x [1] fac(s(x), y) -> fac(p(s(x)), times(s(x), y)) [1] factorial(x) -> fac(x, s(0)) [1] p(v0) -> null_p [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] fac(v0, v1) -> null_fac [0] The TRS has the following type information: plus :: 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac 0 :: 0:s:null_p:null_plus:null_times:null_fac s :: 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac p :: 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac times :: 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac fac :: 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac factorial :: 0:s:null_p:null_plus:null_times:null_fac -> 0:s:null_p:null_plus:null_times:null_fac null_p :: 0:s:null_p:null_plus:null_times:null_fac null_plus :: 0:s:null_p:null_plus:null_times:null_fac null_times :: 0:s:null_p:null_plus:null_times:null_fac null_fac :: 0:s:null_p:null_plus:null_times:null_fac Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_p => 0 null_plus => 0 null_times => 0 null_fac => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: fac(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 fac(z, z') -{ 1 }-> fac(p(1 + x), times(1 + x, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y fac(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 factorial(z) -{ 1 }-> fac(x, 1 + 0) :|: x >= 0, z = x p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(p(1 + x), y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> plus(y, times(p(1 + x), y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) fac(0, z0) -> z0 fac(s(z0), z1) -> fac(p(s(z0)), times(s(z0), z1)) factorial(z0) -> fac(z0, s(0)) Tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(0, z0) -> c2 TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(0)) -> c4 P(s(s(z0))) -> c5(P(s(z0))) FAC(0, z0) -> c6 FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) FACTORIAL(z0) -> c9(FAC(z0, s(0))) S tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(0, z0) -> c2 TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(0)) -> c4 P(s(s(z0))) -> c5(P(s(z0))) FAC(0, z0) -> c6 FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) FACTORIAL(z0) -> c9(FAC(z0, s(0))) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, fac_2, factorial_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_2, FACTORIAL_1 Compound Symbols: c, c1_2, c2, c3_3, c4, c5_1, c6, c7_2, c8_2, c9_1 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FACTORIAL(z0) -> c9(FAC(z0, s(0))) Removed 4 trailing nodes: FAC(0, z0) -> c6 TIMES(0, z0) -> c2 PLUS(0, z0) -> c P(s(0)) -> c4 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) fac(0, z0) -> z0 fac(s(z0), z1) -> fac(p(s(z0)), times(s(z0), z1)) factorial(z0) -> fac(z0, s(0)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, fac_2, factorial_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_2 Compound Symbols: c1_2, c3_3, c5_1, c7_2, c8_2 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: fac(0, z0) -> z0 fac(s(z0), z1) -> fac(p(s(z0)), times(s(z0), z1)) factorial(z0) -> fac(z0, s(0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_2 Compound Symbols: c1_2, c3_3, c5_1, c7_2, c8_2 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) by PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, P_1, FAC_2, PLUS_2 Compound Symbols: c3_3, c5_1, c7_2, c8_2, c1_2 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, P_1, FAC_2, PLUS_2 Compound Symbols: c3_3, c5_1, c7_2, c8_2, c1_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) by TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, FAC_2, PLUS_2, TIMES_2 Compound Symbols: c5_1, c7_2, c8_2, c1_2, c3_3, c3_1 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, FAC_2, PLUS_2, TIMES_2 Compound Symbols: c5_1, c7_2, c8_2, c1_2, c3_3, c3_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(z0), z1) -> c7(FAC(p(s(z0)), times(s(z0), z1)), P(s(z0))) by FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(0), x1) -> c7(FAC(0, times(s(0), x1)), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(0), x1) -> c7(FAC(0, times(s(0), x1)), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(0), x1) -> c7(FAC(0, times(s(0), x1)), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, FAC_2, PLUS_2, TIMES_2 Compound Symbols: c5_1, c8_2, c1_2, c3_3, c3_1, c7_2 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FAC(s(0), x1) -> c7(FAC(0, times(s(0), x1)), P(s(0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, FAC_2, PLUS_2, TIMES_2 Compound Symbols: c5_1, c8_2, c1_2, c3_3, c3_1, c7_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(z0), z1) -> c8(FAC(p(s(z0)), times(s(z0), z1)), TIMES(s(z0), z1)) by FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(0), x1) -> c8(FAC(0, times(s(0), x1)), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(0), x1) -> c8(FAC(0, times(s(0), x1)), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(0), x1) -> c8(FAC(0, times(s(0), x1)), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, PLUS_2, TIMES_2, FAC_2 Compound Symbols: c5_1, c1_2, c3_3, c3_1, c7_2, c8_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, PLUS_2, TIMES_2, FAC_2 Compound Symbols: c5_1, c1_2, c3_3, c3_1, c7_2, c8_2, c8_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(0), x1) -> c8(TIMES(s(0), x1)) We considered the (Usable) Rules:none And the Tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, PLUS_2, TIMES_2, FAC_2 Compound Symbols: c5_1, c1_2, c3_3, c3_1, c7_2, c8_2, c8_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) by PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0)))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0)))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0)))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_3, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_3, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) by TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1, c3_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) by TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1, c3_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(z0), z1) -> c7(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), P(s(z0))) by FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(0, plus(x1, times(p(s(0)), x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(0, plus(x1, times(p(s(0)), x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(0, plus(x1, times(p(s(0)), x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1, c3_3 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FAC(s(0), x1) -> c7(FAC(0, plus(x1, times(p(s(0)), x1))), P(s(0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1))), P(s(0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1, c3_3 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c7_2, c8_2, c8_1, c1_2, c1_1, c3_3, c7_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), times(s(s(z0)), x1)), P(s(s(z0)))) by FAC(s(s(x0)), z1) -> c7(FAC(s(p(s(x0))), plus(z1, times(p(s(s(x0))), z1))), P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) We considered the (Usable) Rules:none And the Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = 0 ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) We considered the (Usable) Rules: p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 And the Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] + x_1 POL(P(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = x_1 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = 0 ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(z0), z1) -> c8(FAC(p(s(z0)), plus(z1, times(p(s(z0)), z1))), TIMES(s(z0), z1)) by FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(FAC(0, plus(x1, times(p(s(0)), x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(FAC(0, plus(x1, times(p(s(0)), x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(FAC(0, plus(x1, times(p(s(0)), x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (83) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) We considered the (Usable) Rules:none And the Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_2, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), times(s(s(z0)), x1)), TIMES(s(s(z0)), x1)) by FAC(s(s(x0)), z1) -> c8(FAC(s(p(s(x0))), plus(z1, times(p(s(s(x0))), z1))), TIMES(s(s(x0)), z1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1, c8_2 ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) We considered the (Usable) Rules:none And the Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = 0 ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1, c8_2 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) We considered the (Usable) Rules: p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 And the Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] + x_1 POL(P(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = x_1 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = 0 ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_1, c1_2, c1_1, c3_3, c7_2, c7_1, c8_2 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) by TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) by TIMES(s(0), z0) -> c3(PLUS(z0, 0)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), x1) -> c7(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), P(s(s(z0)))) by FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: P_1, TIMES_2, FAC_2, PLUS_2 Compound Symbols: c5_1, c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3 ---------------------------------------- (99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(z0))) -> c5(P(s(z0))) by P(s(s(s(y0)))) -> c5(P(s(s(y0)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3, c5_1 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1)), P(s(s(0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3, c5_1 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3, c5_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) by FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3, c5_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(0), x1) -> c7(FAC(p(s(0)), plus(x1, times(0, x1)))) by FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c1_1, c7_2, c7_1, c8_2, c3_3, c5_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) by PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(z0)), x1) -> c7(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), P(s(s(z0)))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c7_1, c8_2, c3_3, c5_1, c1_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(x0)), z1) -> c7(FAC(s(p(s(x0))), plus(z1, times(p(s(s(x0))), z1))), P(s(s(x0)))) by FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c7_1, c8_2, c3_3, c5_1, c1_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(s(z0))), x1) -> c7(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), P(s(s(s(z0))))) by FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c7_1, c8_2, c3_3, c5_1, c1_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(0), x1) -> c8(FAC(p(s(0)), plus(x1, times(0, x1))), TIMES(s(0), x1)) by FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c7_1, c8_2, c3_3, c5_1, c1_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FAC(s(s(x0)), x1) -> c7(P(s(s(x0)))) by FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), x1) -> c8(FAC(p(s(s(z0))), plus(x1, times(s(p(s(z0))), x1))), TIMES(s(s(z0)), x1)) by FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) by FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(z0)), x1) -> c8(FAC(s(p(s(z0))), plus(x1, times(p(s(s(z0))), x1))), TIMES(s(s(z0)), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(x0)), z1) -> c8(FAC(s(p(s(x0))), plus(z1, times(p(s(s(x0))), z1))), TIMES(s(s(x0)), z1)) by FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TIMES(s(0), z0) -> c3(PLUS(z0, 0)) by TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) by FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(s(z0))), x1) -> c8(FAC(s(s(p(s(z0)))), times(s(s(s(z0))), x1)), TIMES(s(s(s(z0))), x1)) by FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(s(y0)))) -> c5(P(s(s(y0)))) by P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c1_1, c5_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), z1) -> c7(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) by FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c1_1, c5_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) by FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c1_1, c5_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) by FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c1_1, c5_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) by PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c5_1, c1_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(0), z0) -> c7(FAC(p(s(0)), plus(z0, 0))) by FAC(s(0), z0) -> c7(FAC(0, plus(z0, 0))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7(FAC(0, plus(z0, 0))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7(FAC(0, plus(z0, 0))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c5_1, c1_1 ---------------------------------------- (143) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7 S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7 K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c5_1, c1_1, c7 ---------------------------------------- (145) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FAC(s(0), z0) -> c7 We considered the (Usable) Rules:none And the Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1, x_2)) = [1] POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7) = 0 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = 0 ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7 S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(0), z0) -> c7 Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c5_1, c1_1, c7 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), P(s(s(z0)))) by FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(0), z0) -> c7 S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(0)), x1) -> c8(FAC(s(0), times(s(s(0)), x1)), TIMES(s(s(0)), x1)) FAC(s(s(0)), x1) -> c7(FAC(s(0), times(s(s(0)), x1))) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(0), z0) -> c7 Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c5_1, c1_1, c7 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FAC(s(0), z0) -> c7 ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c7_1, c5_1, c1_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FAC(s(s(s(y0))), z1) -> c7(P(s(s(s(y0))))) by FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), P(s(s(s(z0))))) by FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(0), z0) -> c8(FAC(p(s(0)), plus(z0, 0)), TIMES(s(0), z0)) by FAC(s(0), z0) -> c8(FAC(0, plus(z0, 0)), TIMES(s(0), z0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(0, plus(z0, 0)), TIMES(s(0), z0)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(FAC(0, plus(z0, 0)), TIMES(s(0), z0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(0), z0) -> c8(TIMES(s(0), z0)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (159) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(0), z0) -> c8(TIMES(s(0), z0)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), z1) -> c8(FAC(p(s(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) by FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) by FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, times(s(p(s(z0))), z1))), TIMES(s(s(z0)), z1)) by FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(p(s(s(0))), z0))), TIMES(s(s(0)), z0)) by FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(s(p(s(0))), z0))), TIMES(s(s(0)), z0)) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(s(p(s(0))), z0))), TIMES(s(s(0)), z0)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(p(s(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) by FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(s(p(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c7_1, c1_1 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(p(s(s(0))), z0)))) by FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(s(p(s(0))), z0)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) Tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) FAC(s(0), x1) -> c8(TIMES(s(0), x1)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(0)), z0) -> c8(FAC(s(0), plus(z0, times(s(p(s(0))), z0))), TIMES(s(s(0)), z0)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) FAC(s(s(0)), z0) -> c7(FAC(s(0), plus(z0, times(s(p(s(0))), z0)))) S tuples: TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) FAC(s(x0), 0) -> c7(FAC(p(s(x0)), times(p(s(x0)), 0)), P(s(x0))) FAC(s(x0), s(z0)) -> c7(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), P(s(x0))) FAC(s(x0), 0) -> c8(FAC(p(s(x0)), times(p(s(x0)), 0)), TIMES(s(x0), 0)) FAC(s(x0), s(z0)) -> c8(FAC(p(s(x0)), s(plus(p(s(z0)), times(p(s(x0)), s(z0))))), TIMES(s(x0), s(z0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) FAC(s(s(z0)), z1) -> c7(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), P(s(s(z0)))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) FAC(s(s(s(z0))), z1) -> c7(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), P(s(s(s(z0))))) FAC(s(s(z0)), z1) -> c8(FAC(s(p(s(z0))), plus(z1, plus(z1, times(p(s(p(s(z0)))), z1)))), TIMES(s(s(z0)), z1)) FAC(s(s(s(z0))), z1) -> c8(FAC(s(s(p(s(z0)))), plus(z1, times(s(p(s(s(z0)))), z1))), TIMES(s(s(s(z0))), z1)) K tuples: FAC(s(0), x1) -> c8(TIMES(s(0), x1)) FAC(s(x0), x1) -> c8(TIMES(s(x0), x1)) FAC(s(s(x0)), x1) -> c8(TIMES(s(s(x0)), x1)) FAC(s(s(s(s(y0)))), z1) -> c7(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2 Defined Pair Symbols: TIMES_2, FAC_2, PLUS_2, P_1 Compound Symbols: c3_1, c8_1, c1_2, c7_2, c8_2, c3_3, c5_1, c1_1, c7_1